HI 


^  Sf-  '^ 


liK 


H 


IN  MEMORIAM 
FLORIAN  CAJORI 


GRAMMAR  SCHOOL  ARITHMETIC 


BT 

BRUCE  M.   WATSON 

SUPERINTENDENT  OF  SCHOOLS,   SPOKANE,  WASH. 
AND 

CHARLES   E.   WHITE 

PEINCIPAL  OF  FRANKLIN   SCHOOL,    SYRACUSE,   N.Y. 


D.    C.    HEATH   &   CO.,   PUBLISHERS 

BOSTON        NEW  YORK        CHICAGO 

1912 


Copyright,  1908, 
By  D.  C.  Heath  &  Co. 


INTRODUCTION 

This  volume,  the  third  of  the  series,  is  designed  for  use  in 
the  higher  grammar   grades. 

It  contains  a  brief  and  somewhat  more  mature  treatment  of 
topics  covered  by  the  first  two  books,  and  a  thorough  course  in 
the  more  advanced  subjects  taught  in  the  upper  classes. 

The  aim  has  been  to  secure  in  pupils  a  high  degree  of  facility 
and  accuracy  in  computation,  to  develop  the  power  of  visual- 
ization, and  to  cultivate  a  habit  of  reliance  upon  independent 
thought,  rather  than  upon  rules  and  formulas,  in  obtaining 
results. 

The  attention  of  teachers  is  especially  directed  to  the  plan  of 
developing  the  basal  ideas  of  each  new  topic  by  means  of  oral 
exercises,  thus  insuring  appreciation  of  the  new  idea  in  advance 
of  the  conventional  form  of  computation. 

The  authors  desire  to  acknowledge  their  obligation  to  Mr. 
Edward  South  worth.  Head  Master  of  the  Mather  School,  Boston, 
Mass.,  for  helpful  suggestions  in  the  treatment  of  Mensuration  ; 
to  Principal  C.  S.  Gibson  and  Miss  Mary  Losacker  of  Seymour 
School,  Syracuse,  N.Y.,  for  aid  in  the  preparation  of  the  sec- 
tions pertaining  to  Interest ;  and  to  the  many  superintendents 
and  others  who  have  performed  a  most  valuable  service  in  read- 
ing and  correcting  the  proofs. 


iii 


TABLE   OF   CONTENTS 

PA6B 

Arabic  Notation  and  Numeration 3 

Roman  Notation 8 

Addition  of  Integers  and  Decimals 10 

Subtraction  of  Integers  and  Decimals 12 

Addition  and  Subtraction  —  Oral .14 

Multiplication  of  Integers 15 

Division  of  Integers 17 

Multiplication  and  Division  of  Decimals 19 

Indicated  Operations 23 

Tests  of  Divisibility 25 

Ideas  of  Proportion 27 

Factors  and  Multiples 29 

Cancellation 31 

Least  Common  Multiple 32 

Greatest  Common  Divisor 34 

Fractions 36 

Reduction  to  Lowest  Terms 37 

Reduction  of  Improper  Fractions  to  Integers  or  Mixed  Numbers  39 

Reduction  of  Integers  and  Mixed  Numbers  to  Improper  Fractions  40 

Reduction  of  Fractions  to  Least  Common  Denominator          .        .  41 

Addition  of  Fractions  and  Mixed  Numbers 42 

Subtraction  of  Fractions  and  Mixed  Numbers        ....  43 

Multiplication  and  Division  Combined 44 

Multiplication .45 

Division 48 

Comparative  Study  of  Decimals  and  Common  Fractions          .  51 

Aliquot  Parts       . 55 

Special  Cases  in  Multiplication     . 57 

Special  Cases  in  Division 58 

Accounts  and  Bills 61 

Review  and  Practice .        .68 

Articles  sold  by  the  Thousand,  Hundred,  or  Hundredweight  75 

Denominate  Numbers 76 

Reduction  of  Denominate  Numbers         . 85 

Addition  and  Subtraction  of  Compound  Numbers  ....  91 

Multiplication  and  Division  of  Compound  Numbers       ...  94 

Measurements       .        . 96 

Areas  of  Parallelograms 96 

Areas  of  Triangles .         .98 

Measurement  of  Rectangular  Solids 100 

Building  Walls 102 

Floor  Covering 104 

Plastering 105 

Wall  Coverings 107 

Lumber  Measure •        •        .        .  108 

iv 


TABLE   OF   CONTENTS  V 

PAGE 

Estimating  Shingles 112 

Volume  and  Capacity 113 

Review  and  Practice 117 

Computation  in  Hundredths 125 

Percentage 128 

Per  Cents  Equivalent  to  Common  Fractions 133 

Profit  and  Loss 139 

Commission 145 

Commercial  Discount 150 

Contracts 157 

Insurance 157 

Interest          .        . 165 

Interest  for  Short  Periods 170 

Exact  Interest 171 

Problems  in  Interest 172 

Compound  Interest 180 

Promissory  Notes 182 

Kinds  of  Notes 185 

Indorsement 186 

Maturity 187 

Default  of  Payment 188 

Exercises 188 

Computing  Interest  on  Notes 191 

Partial  Payments > 193 

Review  and  Practice 198 

Banks  and  Banking 205 

Depositing  and  Withdrawing  Money       .        .        .        .        .        .  206 

Comparison  of  Checks  and  Notes 209 

Bank  Discount 211 

Protesting  Notes,  Checks,  and  Drafts       .        .        .    '     .        .        .  218 

Taxes '  ....  220 

Exchange 226 

Commercial  Drafts 231 

Exchange  by  Postal  Money  Order 235 

Exchange  by  Express  Money  Order 237 

Exchange  by  Telegraph  Money  Order 238 

Foreign  Exchange 239 

Metric  System 247 

Linear  Measure 247 

Surface  Measure 252 

Land  Measure 255 

Volume  Measure 256 

Capacity  Measure 257 

Measures  of  Weight 259 

Duties 261 

Equations 267 

Review  and  Practice  .        •        •        • 276 


vi  TABLE  OF   CONTENTS 

PAGE 

Stocks .  284 

Bonds • 296 

Ratio 303 

Proportion 306 

Partitive  Proportion 312 

Partnership 314 

Review  and  Practice 317 

Involution 323 

Evolution .        ,  326 

Square  Root 329 

Square  Root  of  a  Decimal 336 

Square  Root  of  a  Common  Fraction 337 

Evolution  by  Factoring 339 

Applications  of  Square  Root 340 

Mensuration          . 344 

Plane  Figures 345 

Areas  of  Regular  Polygons 345 

Areas  of  Trapezoids 346 

Study  of  the  Circle 347 

Solids o 351 

Study  of  Prisms 351 

Study  of  the  Cylinder 352 

Study  of  the  Cone 355 

Study  of  Regular  Pyramids 358 

Study  of  the  Sphere 359 

Similar  Surfaces 361 

Longitude  and  Time   .        .        .        , 367 

Standard  Time 371 

Review  and  Practice ,        •        •        .  373 

APPENDIX 

Cube  Root 392 

Similar  Solids 397 

Methods  of  Computing  Interest 397 

Method  by  Aliquot  Parts ....         c         ....  397 

Bankers' Method 398 

Ordinary  Six  Per  Cent  Method 398 

True  Discount  and  Present  Worth 398 

Suretyship 400 

Compound  Proportion 402 

Government  Lands 405 

Greatest  Common  Divisor  by  Continued  Division      .        .        .  406 

Farmers*  Estimates 407 

Kinds  of  Paper  Money 408 

Multiplication  Table 409 

Compound  Interest  Table         .        •        •        •        •       •        .        .  410 


GRAMMAR  SCHOOL  ARITHMETIC 


GRAMMAR  SCHOOL  ARITHMETIC 


ARABIC  NOTATION  AND  NUMERATION 

1.  That  which  tells  how  many  is  number;  e.g.  three,  seven, 
five,  two  and  one  half. 

2.  One  is  a  unit;  e.g.  one,  one  (doHar),  one  (book). 

3.  A  number  that  is  applied  to  some  particular  thing  or  things 
is  caUed  a  concrete  number ;  e.g.  five  (books),  seven  (doHars), 
ten  (months). 

4.  A  number  that  is  not  applied  to  any  particular  thing  or 
things  is  called  an  abstract  number;   e.g.  five,  seven,  eleven. 

To  the  Teacher.  —  See  Primary  Arithmetic,  pages  139  and  140. 

5.  A  number  that  is  composed  entirely  of  whole  units  is  an 
integer;  e.g.  six,  eight,  thirteen. 

6.  One  or  more  of  the  equal  parts  of  a  unit  is  a  fraction;  e.g. 


7     _2_5_ 
?'    10  0- 


7.  The  number  above  the  line  in  a  fraction  is  the  numerator ; 
the  number  below  the  line  in  a  fraction  is  the  denominator ;  e.g. 
in  the  fractions  |-,  |-,  and  -f-^-^.,  the  numerators  are  2,  7,  and  25. 
The  denominators  are  3,  8,  and  100. 

8 .  The  product  of  equal  factors  is  a  power.     (See  §  47) ;  e.g. 

4  is  a  power  of  2    because  2x2=4 

8  is  a  power  of  2    because         2x2x2    =8 

81  is  a  power  of  3    because  3x3x3x3    =81 

100  is  a  power  of  10  because  10  x  10  =  100 


4  GRAMMAR  SCHOOL   ARITHMETIC 

Name  three  other  powers  of  10. 

9.    A  fraction  whose  denominator  is  10  or  a  power  of  10  is  a, 
decimal  fraction ;  e.g.  -^^,  -^q^,  lotoo'  '^^  '^^•>  -0^38. 

10.  Expressing  numbers  by  means  of  figures  or  letters  is 
notation;    e.g.  32,  XXXII. 

11.  Expressing  numbers  by  means  of  figures  is  Arabic  notation; 
e.g.  349,  6872.351. 

1,  2,  3,  4,  5,  6,  7,  8,  and  9  are  called  significant  figures  because 
they  have  values.  The  figure  0,  called  a  cipher,  naught,  or  zero, 
expresses  no  value.  It  is  used  to  give  the  significant  figures 
their  proper  places  in  expressing  numbers. 

12.  The  value  of  each  significant  figure  depends  upon  the 
place  which  it  occupies  when  used  with  other  figures  in  express- 
ing a  number. 

The  value  of  a  figure  in  any  place  is  ten  times  as  great  as  it 
would  be  if  it  occupied  the  next  place  to  the  rights  and  one 
tenth  as  great  as  it  would  be  if  it  occupied  the  next  place  to 
the  left. 

Since  the  value  of  a  figure  is  increased  tenfold  as  it  is  moved 
one  place  from  right  to  left,  and  divided  by  ten  as  it  is  moved 
one  place  from  left  to  right,  Arabic  notation  is  said  to  be  based 
on  a  scale  of  ten  ;  or,  the  scale  of  Arabic  notation  is  a  decimal  scale. 

The  decimal  scale  extends  through  decimal  fractions  as  well 
as  integers,  the  scale  of  increase  and  decrease  being  uniform 
from  the  highest  unit  of  the  integer  to  the  lowest  unit  of  the 
decimal. 

The  names  of  the  units  occupying  the  different  places  are 
called  the  different  orders  of  units;  and  each  group  of  three 
orders  of  units  constitutes  a  period. 

The  left  hand  period  of  an  integer  may  contain  only  one  or  two 
figures,  or  orders  of  units  ;  it  is  then  called  an  incomplete  period. 


ARABIC   NOTATION   AND  NUMERATION 


13.  TABLE   OF   ARABIC  NOTATION 

o  -S  o  "I 

•n  .2  -z  °-  -a 


J  =  o  ^ 

=  •-  ^  c 

CO  2  I-  3 


•C  TJ 


i^  «  U>  «»  5 


«2  E  fs      .  £     rt 

C  -  M        CA  jj  -M        (O 


fe         =  •-  =  ^       Ili  = 

q:=«>  =o»  =«  25  "5  w^tjii 

w-Vo  ■?§  TO  r!2-D«)  ^  -^    -a     »>    V    f> 

w       a)=c<i>=<o<i)=I2©2"Jo»  rt«)Q)coOo+; 

S         T3*r.2x!-?5T3VO-0-»;'3"Dtf)W.E5-0       3r-00 

o       ccEEE^^CEEccocc.-tioccocciE 

^  3a>i-3<Ur=30--3Q)£3<UE0a>3£©3-- 

i|-I-ihcqxhSxi-i-3:i-3Q|-xi-I-i2 
12  7,  34  6,  20  8,  63  5,  409    .    239   107 

Observe  in  the  above  table  that 

a.  The  decimal  point  (.)  is  placed  between  units'  and  tenths' 
places.  Figures  at  the  left  of  the  decimal  point  express  in- 
tegers^ and  figures  at  the  right  of  the  decimal  point  express 
decimal  fractions. 

h.  The  different  orders  of  units  are  numbered  from  the  deci- 
mal point  both  to  the  right  and  to  the  left. 

c.  The  values  of  the  different  orders  of  units  increase  uni- 
formly from  right  to  left  and  decrease  uniformly  from  left  to 
right  in  a  tenfold  ratio,  throughout  the  integer  and  the  decimal. 

d.  The  name  of  each  period  is  the  same  as  that  of  the  riglit- 
hand  place  in  that  period. 

e.  Commas  are  used  to  separate  the  periods,  for  convenience 
in  reading. 

14.  A  number  that  is  composed  of  an  integer  and  a  decimal  is 
called  a  mixed  decimal;  e.g.  2.5,  31.242,  600.00006. 


6  GRAMMAR   SCHOOL   ARITHMETIC 

15.  Naming  the  places  of  figures  and  reading  numbers  is  nu- 
meration ;  e.g.^  to  numerate  the  number  .40236,  we  should  say, 
tenths,  hundredths,  thousandths,  ten-thousandths,  hundred- 
thousandths —  forty  thousand  two  hundred  thirty-six,  hun- 
dred-thousandths. 

16.  In  reading  numbers,  the  word  and  should  not  be  used 
except  between  the  integer  and  the  decimal  of  a  mixed  decimal, 
or  between  the  integer  and  the  fraction  of  a  mixed  number  ;  e.g. 
30,245  is  read,  thirty  thousand  two  hundred  forty -five  ;  .328  is 
read,  three  hundred  twenty-eight  thousandths;  30,245.328 
is  read,  thirty  thousand  two  hundred  forty-five  and  three  hun- 
dred twenty-eight  thousandths. 

17.  Read  the  following  integers  and  write  them  in  words: 


1. 

42,930 

6. 

8,034,034 

11. 

400,000,040 

2. 

80,765 

7. 

3,001,001 

12. 

6,097,429 

3. 

49,060 

8. 

9,705,010 

13. 

913,074,060,812 

4. 

305,041 

9. 

389,046 

14. 

3,501,230,780,020 

5. 

200,030 

10. 

8,107,010 

15. 

600,400,300,001 

18.  Bead  the  following  decimals  and  write  them  in  words 

1.  .34  6.    .8070  *      11.    .20456 

2.  .751        7.  .24305       12.  .380751 

3.  .03         8.  .9280        13.  .0007 

4.  .705        9.  .60834    •  14.  .000007 

5.  .807       10.  .90307       15.  .603120 


ARABIC   NOTATION   AND  NUMERATION  7 

19.    Read   the  following   mixed   decimals  and  write   them  in 


words . 

; 

1. 

64.85 

11. 

9,500.5050 

21. 

900.900 

2. 

289.9 

12. 

384.20108 

22. 

.990 

3. 

407.07 

13. 

70,903.60050 

23. 

6.00006 

4. 

897.403 

14. 

8,000.800 

24. 

•   42.0402 

5. 

2,025.025 

15. 

8,000.00008 

25. 

100.00001 

6. 

83.0008 

16. 

.08008 

26. 

100.100 

7. 

4,920.0020 

17. 

.060010 

27. 

101.101 

8. 

370.0700 

18. 

.0010 

28. 

101.100 

9. 

9,876.540 

19. 

.1010 

29. 

10,010.1010 

10. 

300.00003 

20. 

400,004.00004 

30. 

100,000.100 

20.  Express  the  following  numbers  in  fig\ires: 

1.  Two  hundred  thousand,  two  hundred. 

2.  Twelve  thousand,  and  two  thousandths. 

3.  Eighty-eight  thousand,  and  three  hundredths. 

4.  One  hundred,  and  one  hundred  thousandths. 

5.  One  hundred  thousand,  and  one  hundred-thousandth. 

6.  Three  thousand  one  hundred-thousandths. 

7.  Eight  thousand,  and  eight  thousandths. 

8.  Five  billion,  sixty  thousand,  two  hundred. 

9.  Three  hundred  six  million  six. 

10.  Forty-eight  thousand  two  hundred,  and  two  hundred- 
thousandths. 

11.  Three   hundred   seventy-five   thousand  sixty,  and   four 
hundred  ten  thousandths. 


8  GRAMMAR   SCHOOL   ARITHMETIC 

12.  Seventy  thousand  four  hundred,  and  four  hundred  ten- 
thousandths. 

13.  Sixty  thousand  fifty,  and  sixty-nine  ten-thousandths. 

14.  Ninety-one,  and  ninety-one  thousandths. 

15.  Two  thousand  three  hundred  one,  hundred-thousandths. 

16.  Five  hundred  eighteen,  and  five  hundred  eighteen  ten- 
thousandths. 

17.  Thirty-nine  thousand  four  millionths. 

18.  Two  hundred  two  thousandths. 

19.  Two  hundred,  and  two  thousandths. 

20.  Two  and  two  hundred  thousandths. 

21.  Two  and  two  hundred-thousandths. 

22.  Six  hundred  six  thousand. 

23.  Six  hundred  six  thousandths. 

24.  Six  hundred,  and  six  thousandths. 

25.  Six  hundred,  and  six  hundred  thousandths. 

26.  Six  hundred,  and  six  hundred-thousands. 

ROMAN  NOTATION 

21.    Expressing  numbers  hy  7neans  of  letters  is  Roman  notation. 

For  many  years  the  Roman  system  of  notation  was  commonly  used  in 
Europe.  The  ancient  Greeks  also  had  a  system  of  notation  which  employed 
the  letters  of  the  Greek  alphabet.  Both  of  these  systems  were  awkward, 
and  of  little  use  in  making  computations. 

The  Arabic  numerals  were  used  first  in  India.  The  figure  0  was  lacking 
until  about  the  fifth  century.  Its  introduction  added  greatly  to  the  useful- 
ness of  the  system. 

Arabic  notation  was  first  used  in  Europe  about  the  twelfth  century,  hav- 
ing been  brought  there  by  the  Arabs.  It  is  now  the  prevailing  system  of 
notation  throughout  the  civilized  world. 


ROMAN  NOTATION 


9 


22.  The  Roman  system  of  notation  employs  the  following 
seven  capital  letters  in  expressing  numbers: 

I  (1),  V  (5),  X  (10),  L  (50),  C  (100),  D  (500),  M  (1000). 

In  combining  these  letters,  the  following  principles  are 
observed : 

a.  Repeating  a  letter  repeats  its  value  ;  e.g. 

X  =  10,   XX  =  20,   XXX  =  30. 

h.  When  a  letter  folloivs  one  of  greater  value,  its  value  is  added 
to  the  greater  value  ;  e.g.  C  =  100,  L  =  5jO,  CL  =  150. 

c.  When  a  letter  precedes  one  of  greater  value.,  its  value  is  sub- 
tracted from  the  greater  value  ;  e.g.   C  =  100,  X  =  10,  XC  =  90. 

d.  When  a  letter  is  placed  between  two  letters  of  greater  value, 
its  value  is  subtracted  from  the  sum  of  the  two  greater  values ; 
e.g.  C  =  100,  X  =  10,  L  =  50,  CXL  =  140. 

e.  A  bar  placed  over  a  letter  multiplies  its  value  by  1000  ;  e.g. 
XC  =  90,  XC  =  90,000. 


23.    Read  the  following  numbers  and  express  them  in  Arabic 
numerals: 


1.  IX 

2.  XIII 

3.  XIX 

4.  ccc 

5.  CDVII 

6.  XCVI 

7.  CLXIX 


8.    MDCC 


9.    XVI 

10.  MCMIX 

11.  MCMXI 

12.  QLIII 

13.  M 

14.  CMIX 


15.  DCXL 

16.  LXXyill 

17.  XCV 

18.  XCIV 

19.  CCXCI 

20.  DLXXjq 

21.  MCMXIX 


10  GRAMMAR   SCHOOL   ARITHMETIC 

24.  Express  the  following  numbers  in  Roman  numerals : 

1.  8  5.  86  9.  83  13.  64  17.  237 

2.  18  6.  44  10.  99  14.  110  18.  550 

3.  119  7.  55  11.  14  15.  208  19.  1555 

4.  29  8.  136  12.  75  16.  400  20.  1911 

ADDITION 

25.  Addition  is  the  process  of  uniting  two  or  more  numbers  into 
one  number ;  e.g.  2  +  5=7. 

26.  The  numbers  added  are  addends  ;  e.g.  3  + 10  =  13  ;  3 
and  10  are  the  addends. 

27.  The  result  of  addition  is  the  sum  ;  e.g.  8  books  and  7 
books  are  15  books  ;  15  is  the  sum. 

28.  The  addends  and  the  sum  are  called  the  terms  of  addition. 

29.  The  sign  +  indicates  addition  and  is  read  plus. 

30.  The  sign  = ,  called  the  sign  of  equality.,  is  read  equals.,  and 
indicates  that  the  expression  preceding  it  has  the  same  value 
as  the  expression  following  it. 

31.  In  column  addition,  we  should  learn  to  read  a  column  of 
figures,  catching  the  combinations  of  two  figures  at  a  glance, 
just  as  we  read  a  book  without  stopping  to  spell  the  words. 

In  the  following  examples,  add  by  combinations  of  two  figures 
as  indicated  in  the  units'  column  of  example  1,  especially  where 
the  sum  of  two  figures  does  not  exceed  12. 

Test  each  sum  by  adding  in  reverse  order.  Time  yourself 
and  see  how  quickly  you  can  get  correct  answers. 


ADDITION 

1. 

2. 

3. 

4. 

5. 

2351 
682  j 

838 

925 

28 

89 

209 

7463 

39 

22 

796  J 

761 

729 

476 

917 

489 

5834 

834 

483 

899 

117 

609 

276 

9876 

4681 

1ft 

343 

182 

9013 

2345 

722, 

flO 

536 

4231 

1862 

1076 

326|^^ 
245  J 

498 

5687 

918 

8864 

704 

21 

4705 

173 

348/ 

428 

4598 

8196 

8888 

193 

234 

729 

2222 

11 


32.  Numbers  to  he  added  should  he  written  so  that  units  of 
the  same  order  stand  in  the  same  column. 

In  writing  decimals,  this  will  be  accomplished  by  making  the  decimal 
points  stand  in  a  column. 

Dimes,  cents,  and  mills  are  expressed  decimally  as  tenths,  hundredths, 
and  thousandths  of  a  dollar. 

33.  In  examples  1-5,  add  and  test,  timing  yourself. 


1. 

2. 

3. 

4. 

5. 

1  34.25 

1  3.82 

1  9.764 

i  48.39 

1186.424 

69.87 

14.32 

5.20 

446.19 

4.2468 

801.06 

90.125 

49.0742 

72.934 

.9374 

12.14 

6.187 

.894 

693.126 

102.0738 

198.28 

2.353 

2.763 

28.987 

84.176 

79.63 

4.589 

.058 

6.104 

9.334 

918.47 

81.236 

.9278 

92.193 

19.2376 

29.13 

9.88 

4.615 

8.56 

5.28 

40.88 

71.24 

.8923 

.79 

80.342 

60.82 

3.257 

.705 

42.138 

9.76 

41.98 

4.934 

400.0006 

8.973 

3.582 

12  GRAMMAR  SCHOOL   ARITHMETIC 

6.  Add  four  dollars  and  ninety-one  cents,  sixty-three  dollars 
seventy-five  cents  and  eight  mills,  twenty-seven  dollars  forty- 
two  cents  and  two  mills,  three  hundred  seventy-eight  dollars 
twenty-nine  cents  and  seven  mills,  nine  hundred  forty-two  dol- 
lars, six  dollars  and  seventy-eight  cents. 

7.  Find  the  sum  of  eighty-one  and  eighty-one  thousandths, 
sixty-three  and  twenty-nine  hundredths,  two  hundred  fourteen 
and  one  hundred  fifty-eight  ten-thousandths,  five  hundred  six- 
teen thousandths,  twenty-nine  and  forty-four  ten-thousandths, 
six  hundred  eighty-four  ten-thousandths,  ninety -six  ten- 
thousandths,  fifty-six  ten-thousandths,  seventy-six  and  eight 
tenths. 

SUBTRACTION 

34.  Subtraction  is  the  process  of  finding  the  difference  between 
two  numbers ;  eg,  21  —  7  =  14  ;   13  cents  —  5  cents  =  8  cents. 

35.  The  number  from  which  we  subtract  is  the  minuend.  The 
number  subtracted  is  the  subtrahend.  The  result  of  subtraction 
is. the  difference  or  remainder. 

36.  The  difference  is  always  the  number  that  must  he  added 
to  the  subtrahend  to  obtain  the  minuend  ;  e.g.  17  —  9  =  8.  17 
is  the  minuend,  9  is  the  subtrahend,  and  8  is  the  difference  or 
remainder. 

37.  The  minuend^  subtrahend^  and  remainder  are  the  terms 
of  subtraction. 

38.  The  sign  —  indicates  subtraction  and  is  read  minus,  or 


39.  Numbers  to  be  subtracted  should  be  written  so  that 
units  of  any  order  in  the  subtrahend  stand  under  units  of  the 
same  order  in  the  minuend. 


SUBTRACTION 


13 


40.  The  correctness  of  work  in  subtraction  may  be  tested  by 
adding  the  remainder  and  the  subtrahend.  If  this  gives  the 
minuend,  the  work  is  correct. 

41.  1.    From  7364  take  3875. 


Since  5  units  cannot  be  taken  from  4  units,  we  take  1  ten  from 
6  tens,  which,  united  with  4  units,  makes  14  units.  5  tens  remain  in 
the  minuend.     5  units  from  14  units  leave  9  units. 

In  a  similar  manner,  7  tens  from  15  tens  leave  8  tens,  8  hundreds 
from  12  hundreds  leave  4  hundreds,  and  3  thousands  from  6  thousands 
leave  3  thousands. 

The  entire  remainder  is  3489. 


7364 

3875 
3489 


Subtract  and  test,  timing  yourself: 


2. 

3042 

.  825 

3. 

63895 

4287 

4. 

2961 
1953 

5. 

27409 
8129 

6. 

63204 
9183 

7. 

28654 
17946 

8. 

10090 
1095 

9. 

26130 
9231 

10. 

31024 
2736 

11. 

82431 
8243 

12. 

63205 
8164 

13. 

90372 
82365 

14. 

64351 

27809- 

15. 

30756 
7912 

16. 

12005 
11996 

17. 

3102 
471 

18. 

28143 
9204 

19. 

2000 
199 

20. 

7000 
6999 

21. 

202 
193 

Decimals  to  be  subtracted  should  be  written  so  that  the  deci- 
mal point  of  the  subtrahend  comes  directly  below  that  of  the 
minuend.     Why? 

When  the  subtrahend  contains  more  decimal  places  than  the 
minuend,  we  may  subtract  as  though  ciphers  were  annexed  to 


14  GRAMMAR   SCHOOL   ARITHMETIC 

the  minuend  to  make  as  many  decimal  places  in  the  minuend 
as  in  the  subtrahend. 

Annexing  ciphers  to  a  decimal  affects  its  value  how?    Why? 

42.    Written 
1.    Subtract  39.2479  from  167.3. 

167.3 
39.2479 


128.0521     Difference  or  Remainder 

Find  the  remainders^  and  test  without  re-writing  the  numbers 

2.  43527-389.19  12.    384.79-93.6215 

3.  168.42  -  $42.93  13.    29.810  -  13.7901 

4.  $365.-846.12  14.    6.8001-5.80013 

5.  $89.10-123.562  15.    $40.78 -$29,783 

6.  30.-4.7619  16.   8050.706-805,0706 

7.  563. -.9999  17.   423.7-42.37 

8.  $913. -$.258  18.    68023.4-234 

9.  63.9542-18.156  19.   $121.075 -$12.10 

10.  864.23-1.0009  20.    76513.28-7762.47103 

11.  909.091-89.0902  21.    83.54 -.7592 

ADDITION  AND  SUBTRACTION 
43.    Oral 

1.    Add  84  and  79  2.    Subtract  76  from  91 

84  H-  70  =  154  91  -  70  =  21 

154+    9  =  163  21-6  =  15 

Say  84,  154,  163.  Say  91,  21,  15. 


MULTIPLICATION  15 

3.  35  +  19  9.  29  +  34  15.  57  +  24 

4.  46  +  15  10.  83-47  16.  .f.86-$.38 

5.  $.83-f.l4  11.  f.79  +  1.24  17.  1.79  +  1.42 

6.  $.37  +  1.48  12.  $1.21  +  1.43  18.  $1.20-1.84 

7.  $.92 -$.25  13.  115-37  19.  $2.80 +  $.70 

8.  88-39  14.  36  +  45  20.  $1.50-$. 73 

MULTIPLICATION 

44.  Multiplication  is  taking  one  number  as  many  times  as  there 
are  units  in  another ;  e.g,  6  times  9  =  54. 

45.  The  number  multiplied  \^  the  multiplicand  ;  the  number  by 
which  we  multiply  is  the  multiplier  ;  the  result  of  multiplication 
is  the  product ;  e.g,  12  times  20  are  240.  20  is  the  multiplicand^ 
12  is  the  multiplier^  and  240  is  the  product,  20  and  12  are 
factors  of  240. 

46.  The  multiplier^  multiplicand^  and  product  are  the  terms 
of  multiplication. 

47.  Uach  of  the  numbers  that  are  multiplied  to  produce  a  num- 
ber is  a  factor  of  that  number ;  e.g,  2,  3,  and  5  are  factors  of 
30  because  2  x  3  x  5  =  30. 

48.  The  multiplier  and  multiplicand  are  factors  of  the  product. 
The  product  is  the  same  in  whatever  order  the  factors  are 
taken ;  e.g.  6  times  7  =  42,  and  7  times  6  =  42 ;  3  x  5  x  4  =  60, 
and  4  X  3  X  5  =  60. 

49.  The  sign  x ,  placed  between  two  numbers,  indicates  that 
one  of  them  is  to  be  multiplied  by  the  other. 


16 


GRAMMAR  SCHOOL  ARITHMETIC 


50.  Multiplication  of  integers  is  a  short  method  of  adding 
equal  integers  ;  e.g.  4x5  may  be  obtained  by  adding  four  5's, 
thus  5  +  5  +  5  +  5  =  20,  or  by  adding  five  4's,  thus  4  +  4  +  4 
+  4  +  4  =  20. 

51.  Oral 
1. 


3 

8 

8 
12 

5 

6 

7 
2 

1 

9 

0 

6 

11 

4 

10 

12 

2 

11 

5 

7 

10 

3 

1 

4 

9 

0 

Multiply  each  number  in  the  upper  row  by  every  number  in 
the  lower  row. 

2.  What  is  the  effect  of  annexing  a  cipher  to  an  integer  ? 
Two  ciphers?    Three  ciphers ?    Four  ciphers  ? 

3.  How  may  we  multiply  a  number  by  100,000  ? 

4.  300  =  3  X  100     15  X  300  =  15  X  3  X  100  =  ? 

5.  Multiply  25  by  10 ;  by  100  ;  by  1000. 

6.  Multiply  368  by  10  ;  by  100  ;   by  1000. 

7.  Multiply  9  by  10,000  ;  by  100,000  ;  by  1,000,000. 

8.  Multiply  12  by  40 ;  by  400  ;  by  4000. 

9.  Multiply  36  by  200 ;  by  2000 ;  by  20,000. 

10.    Multiply  70  by  1000 ;  by  50 ;  by  800  ;  by  4000. 

52.    Written 
1.    Multiply  5972  by  689. 

In  multiplying  5972  by  689  we  multiply  by  9,  by  80,  and 

^""      by  600,  and  add  the  results  (called  partial  products).     The 

53748      sum  of  the  pa^rtial  products  is  the  product  required. 

47776  ^6  omit  the  ciphers  at  the  right  of  the  partial  products 

S5882  after  the  first.     The  second  partial  product  is  477,760.     Read 

A1  IzlTHft      ^^  third  partial  product. 


DIVISION  17 

2.    Multiply  864  by  403. 
864 

403  When  the  multiplier  contains  a  cipher,  a  partial  product 

2592      is  omitted.     Why?      The  next  partial    product    begins   two 
3456  places  to  the  left.     Read  the  partial  products  in  example  2. 

348192 

In  examples  3-17,  find  the  products^    and  read  each  partial 
product : 

3.  368x29  8.  7359x83  13.  4907x199 

4.  4596  X  38  9.  9138  x  43  14.  2051  x  7892 

5.  6874  X  63  10.  294  x  137  15.  345  x  4006 

6.  1024x99  11.  809x809  16.  4239x618 

7.  2809  X  83  12.  799  x  835  17.  9999  x  8507 

DIVISION 

53.  Division  is  the  process  of  finding  one  of  two  factors^  when 
the  other  factor  and  the  product  are  given  ;  e.g.  35  is  the  prod- 
uct of  7  and  5  ;  35  divided  by  7  equals  5  ;  35  divided  by  5 
equals   7. 

54.  The  number  divided  is  the  dividend. 

55.  The  number  by  which  we  divide  is  the  divisor. 

56.  The  number  obtained  by  division  is  the  quotient. 

57.  When  the  divisor  is  not  exactly  contained  in  the  dividend^ 
the  part  of  the  dividend  that  is  left  is  called  the  remainder. 

Name  the  terms  used  in  division. 

58.  The  sign  -?-  between  two  numbers  indicates  that  the  first 
is  to  be  divided  by  the  second.     Division  may  also  be  indicated 


18  GRAMMAR  SCHOOL   ARITHMETIC 

by  writing  the  dividend  above,  and  the  divisor  below,  a  hori- 
zontal line  ;   e.g. 

35  -!-    7  or  -3y5.  means  35  divided  by  7. 
3  -J- 11  or  Y^j  means    3  divided  by  11. 

59.  Division  may  be  tested  by  multiplying  the  divisor  and 
quotient  together  and  adding  the  remainder,  if  there  is  one. 
If  this  result  equals  the  dividend,  the  work  is  correct.     Why  ? 

60.  1.    Divide  981,504  by  213. 

4608   Quotient 

213)981504  213  is  not  contained  in  9  or  98,  but  is  contained  in  981 

852  four  times.    This  is  4  thousands  because  981  is  thousands, 

J295  The  remainder  is  129   (thousands).     Bringing  down  5 

1278  (hundred),  we  have  1295  (hundred),  which  contains  213 

yjCiA      six  (hundred)  times,  with  a  remainder  of  17  (hundred). 

y7QA      Bringing  down  the  cipher,  we  have  170  (tens),  which 

does  not  contain  213  any  tens  times.      We  write  0  in 

tens'  place  in  the  quotient,  bring  down  4,  and  obtain  8  units  for  the  last 
figure  of  the  quotient,  with  no  remainder. 

Note  1.  —  In  the  above  example,  we  may  obtain  the  quotient  figures  by 
using  the  left-hand  figure  of  the  divisor  for  a  guide  figure,  thus,  2  in  9,  four 
times ;  2  in  12,  six  times ;  2  in  17,  eight  times. 

When  the  second  figure  of  the  divisor  is  7,  8,  or  9,  we  may  add  1  to  the 
left-hand  figure  for  a  guide  figure;  e.g.  if  the  divisor  is  286,  it  is  nearly 
300 ;  therefore,  we  may  use  3  for  a  guide  figure  instead  of  2.  When  the 
second  figure  of  the  divisor  is  5  or  6,  we  may  take  for  the  guide  figures  both 
the  left-hand  figure  and  the  left-hand  figure  plus  1. 

Note  2.  —  When  the  divisor  is  not  greater  than  12,  the  quotient  should 
be  obtained  by  short  division ;  that  is,  by  expressing  only  the  dividend,  divisor, 
and  quotient.  The  quotient  may  then  be  placed  either  above  or  below  the 
dividend,  according  to  convenience,  thus, 

Divisor  12)564677       Dividend  47056^5     Quotient 

47056^2   Quotient  12)564677       Dividend 


MULTIPLICATION  AND  DIVISION  OF  DECIMALS         19 

2.  2785-^5  5.    2796-6  8.    68,347^12 

3.  3928^6  6.    61,933^9  9.    7,640,328-^-12 

4.  2890-7  7.    137,401-11  lo.    29,346^11 

11.  3249-^10  21.  912,946^24 

12.  32,695  H- 57  22.  427,473^97 

13.  33,874-^49  23.  9664-16 

14.  99,003^25  24.  13,734-18 

15.  45,914-59  25.  62,826^74 

16.  335,630-62  26.  2,098,119-987 

17.  491,289  H- 73  27.  67,117,890-98 

18.  216,428-^84  28.  29,067,642^1032 

19.  412,582-58  29.  65,980,064^5004 

20.  981,384-75 

30.  What  number  multiplied  by  351  will  give  347,692  for 
a  product? 

31.  1,993,164  is  the  product  of  489  and  what  other  number? 

32.  By  what  must  982  be  multiplied  to  obtain  3,537,492  ? 

MULTIPLICATION  AND  DIVISION  OF  DECIMALS 
61.    Oral 

1.  Moving  a  figure  one  place  to  the  right  affects  its  value 
how?     Two  places?     Three  places?     Four  places? 

2.  Pointing  off  one  decimal  place  in  a  number  is  the  same  as 
moving  all  the  figures  of  the  number  one  place  to  the  right. 
How  does  it  affect  the  value  of  the  number  ? 


20  GRAMMAR   SCHOOL   ARITHMETIC 

3.  Pointing  off  two  decimal  places  in  a  number  affects  its 
value  how  ?     Three  places  ?     Four  places  ? 

4.  How  many  decimal  places  must  we  point  off  in  a  number 
to  divide  it  by  10  ?  by  1000  ?  by  100  ?  by  10000? 

5.  Divide  12,468  by  10  ;  by  100  ;  by  1000  ;  by  10,000. 

6.  Divide  367.54  by  10  ;  by  100  ;  by  1000;  by  10,000. 

7.  How  may  any  integer  be  divided  by  10  ?  by  100  ?  by  1000  ? 

8.  How  may  any  decimal  be  divided  by  10  ?  by  100  ? 
by  1000  ? 

62.     Written 

1.  Multiply  3.456  by  2.47. 

3  ^^Q  3.456  =  3456  -^  1000 

'  2  47  2.47  =  247  ^  100 

-24192  ^•^''^^  X  2.47  =  3456  x  247  -f- 1000  -f- 100 

-^3g24  ^^^6  X  247  =  853632 

6912  853632  -- 1000  -j- 100  =  8.53632 

TTToaoo  ^^  divide  853,632  by  1000  and  100  by  pointing  off  3  +  2,  or  5, 
^•^'^'^'^^  decimal  places. 

•  Summary 

To  multiply  decimals^  multiply  them  as  integers.  Point  off  in 
the  product  as  many  decimal  places  as  there  are  decimal  places  in 
both  factors.  If  the  number  of  figures  in  the  product  is  less  than 
the  required  number  of  decimal  places,  prefix  ciphers. 


2. 

32.5  X  17 

7. 

4.039  X  .24 

12. 

.4907  X  .018 

3. 

426  X  5.9 

8. 

.875  X  1.9 

13. 

.029  X  568 

4. 

3.08  X  6.7 

9. 

13.55  X  .037 

14. 

2.879  X  .015 

5. 

6.015  X  3.1 

10. 

.068  X  5.81 

15. 

.030  X  5960 

6. 

42,805  X  .6 

11. 

.351  X  .42 

16. 

42.691  X. 08 

20. 

93.50  X  78.92 

23 

.9'J9  X  1000 

21. 

9.10  X  .086 

24. 

.888  X  8.88 

22. 

4.375  X  .092 

25. 

15.15  X  98.07 

MULTIPLICATION   AND   DIVISION   OF   DECIMALS  21 

17.  30.01  X  3.400 

18.  .9756  X  84 

19.  .0231  X  .098 

63.  Oral 

1.  One  factor  lias  three  decimal  places,  the  other  four.     How 
many  has  the  product  ? 

2.  The  product  has  four  decimal  places,  the  multiplicand 
one.     How  many  has  the  multiplier  ? 

3.  The  product  has  six  decimal  places,  the  multiplier  three. 
How  many  has  the  multiplicand  ? 

4.  The  product  has  four  decimal  places.     What  could  be  the 
number  of  decimal  places  in  each  of  the  factors  ? 

64.  Written 

1.    Divide  27.3587  by  4.7. 
5.821 


4.7  27.3587 


The  quotient  and  divisor  are  factors  of  what? 

^^^  The  dividend  is  what  of  the  divisor  and  quotient? 

385  When  the  factors  are  given,  how  may  the  number 

376  of  decimal  places  in  the  product  be  found  ? 

QQ  When  the  product  and  one  factor  are  known,  how 

Q  .  may  the  number  of  decimal  places  in  the  other  factor 

^—  be  found  ? 

47 
47 

Summary 

To  divide  decimals^  divide  as  with  integers  and  point  off  in  the 
quotient  as  many  decimal  places  as  there  are  in  the  dividend, 
minus  the  number  of  decimal  places  in  the  divisor. 

If  the  dividend  contains  fewer  decimal  places  than  the  divisor^ 
annex  ciphers  to  make  the  required  number. 


22  GRAMMAR  SCHOOL   ARITHMETIC 

Note. — It  has  been  found  helpful  to  make  a  dot,  before  dividing,  as 
many  places  to  the  right  of  the  decimal  point  in  the  dividend  as  there  are 
decimal  places  in  the  divisor,  and  on  a  line  with  the  tops  of  the  figures, 
making  the  decimal  point  in  the  quotient  directly  over  this  dot,  thus : 

5.821 


4.7 

27.3-587 

Divide  and  test : 

2.    27.72  by  3.85 

12. 

340.2  by  .042 

3.    5074.65  by  56.7 

13. 

34,177  by  14.3 

4.    10.5252  by  2.94 

14. 

190.0892  by  20.3 

5.    6.79592  by  .76 

15. 

8.19  by  195 

6.    111.34  by  293 

16. 

35.434  by  .014 

7.    16.35  by  .025 

17. 

8674.975  by  .025 

8.    205.3758  by  64.2 

18. 

397  by  .125 

9.    102.6  by  .27 

19. 

273.273  by  63.7 

10.    7644  by  .84 

20. 

1.906438  by  .634 

11.    6793.2  by  .999 

21. 

33.84387  by  3890.1 

65.    Find  the  quotients 

correct  to  three  decimal  places : 

1.  439 -r- 86 

2.  92-407 

8. 

46 

987 

12.     42.8 
639.4 

3.    9.91-13 

32 

n^        99 

4.    8645-237 

9. 

416 

"'•    880 

5.    42.356 -.029 
^     86.924 

10. 

89.1 
190C 

» 

14.     ^^-^1 

760 

^'      .39 

11. 

29.41 

15.    287.5 
4100 

999 

5000 

3025 


INDICATED   OPERATIONS 


23 


INDICATED  OPERATIONS 
66.    The  signs  of  aggregation  are : 

a.   Parentheses  (         )  c.    Brackets 


h.    Braces 


[     ] 


Vinculum 


An  expression  written  within,  or  included  by,  any  of  these 
signs  is  to  be  treated  as  a  single  number. 

67.  The  operations  indicated  within  a  sign  of  aggregation 
must  he  performed  before  those  operations  indicated  outside  the 
sign;  e.g. 

40  X  (9 -6)^  [2 +  4]  = 
40  X      3        -      6      =  20 

68.  When  several  successive  operations  are  indicated  with- 
out the  use  of  signs  of  aggregation,  the  indicated  multiplication 
and  division  must  be  performed  before  the  indicated  addition  and 
subtraction;  e.g. 

40x9-6-^2  +  4  = 
360   -     3     +4  =  361 

69.  Oral 

1.  4  +  3x2  =  ? 

2.  (4+3)x2  =  ? 

3.  4x  3  +  2  =  ? 


10.    9  X  (2 +3)^3  =  ? 


4.  4x(3  +  2)  =  ? 

5.  8  +  4-2  =  ? 

6.  (8  +  4)--2  =  ? 

7.  8x7  +  21-3  =  ? 

8.  (8  +  4)  +  9-^3=? 

9.  9x7-21^3  =  ? 


11.  18^6  +  3  =  ? 

12.  18-^(6  +  3)  =  ? 

13.  11x7-3x2=? 

14.  Ilx(7-3)x2  =  ? 

15.  14^2  +  5x8  =  ? 

16.  14^(2  +  5)  X  8  =? 

17.  36 -r-  9-6  ^3  =  ? 

18.  36^(9-6)-^3  =  ? 


I 


24  GRAMMAR  SCHOOL  ARITHMETIC 

70.     Written 

Perform  the  operations  indicated: 

1.  25.13-^(47.2-43.7) 

2.  2.85  X  [9.6 +  3.02 +  .86] 

3.  2.03  X  607.015 -59.6034 


4.  2.03x607.015-59.6034 

5.  487 +  598 +  {6.45- (20.3- 14.35)} 

6.  41.983 -.87  x  10.3 +  .047 

7.  (41. 983-. 87)  X  [10.3 +  .047] 

8.  2310  -  [10  X. 7] +  604x3.50 


9.    378.34-58.7  +  649.83x64.8-6.48 
71.   Indicate  the  operations  and  solve  : 

1.  The  difference  between  496.37  and  288.037,  multiplied 
by  the  quotient  of  183.75  divided  by  2.5. 

2.  A  grocer  bought  a  load  of  potatoes  containing  48  bushels, 
at  65  cents  a  bushel,  and  sold  them  at  80  cents  a  bushel. 
What  was  his  profit  ? 

3.  The  product  of  three  numbers  is  18.902.  Two  of  the 
numbers  are  .02  and  130.     Find  the  other. 

4.  A  machinist  earns  $1080  a  year.  He  pays  $180  a  year 
for  rent,  $306  for  food,  and  $369  for  other  expenses.  In  how 
many  years,  at  that  rate,  can  he  save  $900? 

5.  What  number  divided  by  20.8  will  give  the  quotient 
85  and  the  remainder  11.7? 

6.  A  city  lot  worth  $1200  and  three  carriages  at  $190  each 
were  given  in  exchange  for  30  acres  of  land.  At  what  price 
per  acre  was  the  land  valued  ? 


TESTS  OF  DIVISIBILITY  25 

7.  A  confectioner  put  2151  pounds  of  candy  into  boxes 
holding  1  pound,  3  pounds,  and  5  pounds,  respectively,  using 
the  same  number  of  boxes  of  each  kind.  How  many  boxes 
were  used  for  all  the  candy  ? 

8.  What  number  must  be  added  to  the  sum  of  342.807, 
231.96,  and  324.7  to  equal  the  difference  between  2107.62  and 
and  1009.006? 

9.  A  certain  number  was  divided,  and  20.45  was  both 
quotient  and  divisor.     What  was  the  number  divided  ? 

10.  Make  and  solve  a  problem  that  requires  the  product  of 
two  numbers  to  be  subtracted  from  the  product  of  two  other 
numbers. 

11.  Make  and  solve  a  problem  that  requires  the  product  of 
two  numbers  to  be  added  to  the  product  of  two  other  numbers 
and  the  sum  divided  by  a  certain  number. 

12.  Divide  by  37  the  result  obtained  by  adding  111  to  the 
product  of  148  and  6090. 

13.  A  merchant  bought  345  pounds  of  wool  of  one  man,  3067 
pounds  of  another,  468  pounds  of  another,  and  384  pounds  of 
another,  and  sold  -|-  of  it  at  27  cents  a  pound.  What  did  he 
receive  for  the  part  sold  ? 

14.  Make  and  solve  a  problem  that  may  be  indicated  thus: 
110-  (f  .35  +  12.20  + 16.19  +  f  .18). 

TESTS  OF  DIVISIBILITY 

72.  The  figures  used  in  Arabic  notation  are  called  digits.  Name 
the  digits. 

73.  A  number  that  can  be  exactly  divided  5y  2  is  an  even  num- 
ber; e.g.  2,4,18. 


26  GRAMMAR  SCHOOL  ARITHMETIC 

74.  A  number  that  cannot  be  exactly  divided  by  2  is  an  odd 
number;  e.g.  3,  7,  19. 

75.  A  number  is  exactly  divisible 

a.  By  2,  if  the  digit  in  units'  place  is  0  or  even;  e.g.  70, 
since  the  units'  digit  is  0;  35,976,  since  the  units'  digit  is 
even. 

b.  By  4,  if  the  digits  in  units''  and  tens''  places  are  O's ;  or  if 
the  number  expressed  by  them  is  divisible  by  4 ;  e.g.  3100,  3976. 
How  do  you  know  ?  How  can  we  tell  without  actual  trial  that 
2398  is  not  divisible  by  4  ? 

c.  By  8,  if  the  digits  in  units',  tens\  and  hundreds'  places  are 
O's,  or  if  the  number  expressed  by  them  is  divisible  by  8 ; 
e.g.  11,000  and  37,112.  How  do  you  know?  Why  not 
76,518? 

d.  By  3,  if  the  sum  of  its  digits  is  divisible  by  3;  e.g.  24,762, 
since  2  +  4  +  7  +  6  +  2,  or  21,  is  divisible  by  3. 

e.  By  9,  if  the  sum  of  its  digits  is  divisible  by  9;  e.g. 
397,647,  since  3  +  9+7+6  +  4+7,  or  36,  is  divisible 
by  9. 

/.    By  5,  if  the  units'  digit  is  0  or  5;  e.g.  80 ;  115. 

g.  By  25,  if  the  units'  and  tens'  figures  are  O's,  or  if  the 
number  expressed  by  them  is  divisible  by  25 ;  e.g.  1900 ; 
8375. 

h.  By  125,  if  the  units',  tens',  and  hundreds'  figures  are  O's, 
or  if  the  number  expressed  by  them  is  divisible  by  125;  e.g.  13,000  ; 
71,750. 

i.  By  10  or  a  power  of  10,  if  it  contains  as  many  O's  at  the 
right  of  its  significant  figures  as  there  are  O's  at  the  right  of  the 
1  in  the  divisor;  e.g.  390  is  divisible  by  10;  390,000  is  divisible 
by  10,000. 

y.  By  6,  if  the  number  is  even  and  the  sum  of  its  digits  is 
divisible  by  3 ;  e.g.  21,106. 


IDEAS  OF  PROPORTION  27 

76.    Oral 

1.    Test  each  of  the  following  numbers  for  divisihility  by  2,  4,  8, 
3,  6,  9,  5,  25,  and  125 : 


a. 

1440 

/. 

22,825 

k. 

108,819 

p.        429,000 

b. 

4950 

9- 

54,901 

I. 

90,626 

q.      6,485,479 

c. 

4875 

h. 

1,629,433 

m. 

35,015 

r.   20,525,750 

d. 

36,090 

i. 

302,275 

n. 

833,950 

s.     9,031,330 

e. 

711,000 

J- 

181,365 

0. 

1,530,000 

t.     1,234,567 

2.  An  even  number  will  not  exactly  divide  an  odd  number. 
Why? 

3.  What  numbers  can  be  exactly  divided  by  6  ?  Can  you 
tell  what  numbers  may  be  exactly  divided  by  18  ? 

4.  If  10  will  divide  a  given  number,  what  other  numbers 
will  divide  the  same  number  ? 

5.  If  8  will  divide  a  given  number,  what  other  numbers  will 
divide  it  ? 

6.  If  125  will  divide  a  given  number,  what  other  numbers 
will  divide  it? 

IDEAS  OF  PROPORTION 

77.    Oral 

1.  36  is  how  many  times  12  ?  12  is  what  part  of  36?  If  12 
oranges  cost  8.35,  36  oranges  will  cost  how  many  times  8.35? 
How  much  will  they  cost?  How  many  oranges  can  be  bought 
for  8.70? 

2.  125  is  what  part  of  500?  If  500  sheets  of  paper  cost  90 
cents,  100  sheets  will  cost  what  part  of  90  cents?  100  sheets 
will  cost  how  much  ?  At  the  same  rate,  how  many  sheets  can 
be  bought  for  9  cents?     For  45  cents? 


28  GRAMMAR   SCHOOL   ARITHMETIC 

3.  A  3-pound  basket  of  grapes  cost  10  cents.  At  the  same 
rate,  what  must  be  paid  for  a  12-pound  basket?  How  many 
pounds  can  be  bought  for  50  cents? 

4.  An  automobile  travels  67  miles  in  4  hours.  At  the  same 
rate,  how  far  will  it  travel  in  8  hours?  In  what  time  will  it 
travel  33i  miles? 

5.  A  Vermont  farmer  made  7  pounds  of  maple  sugar  from 
23  gallons  of  sap.  At  that  rate,  how  many  gallons  of  sap  were 
required  for  35  pounds  of  sugar  ?  How  many  pounds  of  sugar 
could  be  made  from  92  gallons  of  sap  ? 

6.  A  man  is  paid  for  his  work  at  the  rate  of  il7  for  44  hours' 
work.  What  does  he  receive  for  11  hours'  work?  How  long 
must  he  work  to  earn  $S^? 

7.  A  Kansas  farmer  raised  518  bushels  of  wheat  on  14  acres 
of  land.  That  was  an  average  of  how  many  bushels  on  two 
acres?  259  bushels  of  this  crop  were  raised  on  how  many 
acres?  How  many  acres  would  be  required  to  produce  1036 
bushels,  at  the  same  rate  ?  How  many  bushels  could  be  raised 
on  42  acres  at  the  same  rate? 

8.  It  required  1 110  a  week  to  buy  food  for  40  boarders 
at  a  certain  boarding  house.  What  would  be  the  weekly 
cost  of  food  for  160  boarders,  at  the  same  rate?  How 
many  persons  could  be  fed  for  $11  per  week? 

9.  If  300  quarts  of  milk  cost  $  21,  what  will  300  gallons 
cost,  at  the  same  price  per  quart? 

10.  How  many  books  at  32^  each  will  cost  as  much  as  405 
books  at  96^  each  ? 

11.  In  how  many  minutes  will  a  steamer,  going  100  rods  a 
minute,  go  as  far  as  a  man  will  row  in  28  minutes,  if  he  rows 
25  rods  a  minute  ? 


FACTORS  AND  MULTIPLES  29 

FACTORS  AND  MULTIPLES 

78.  A  number  that  exaetly  contains  another  number  is  a  mul- 
tiple of  that  number ;  e.g.  21  is  a  multiple  of  7.  It  is  also  a 
multiple  of  3. 

79.  A  factor  that  is  an  integer  is  called  an  integral  factor  ;  e.g. 
8  is  an  integral  factor  of  b6. 

80.  A  number  that  is  not  the  product  of  integral  factors  other 
than  itself  and  1  is  a  prime  number ;  e.g.  2,  3,  5,  7,  11,  and  13. 

81.  A  number  that  is  the  product  of  integral  factors  other  than 
itself  and  1  is  a  composite  number;  e.g.  16,  24,  35,  1000. 

82.  A  factor  that  is  a  prime  number  is  a  prime  factor;  e.g.  13 
is  a  prime  factor  of  26. 

Note.  — In  finding  the  factors  of  a  number  it  is  customary  to  consider  only- 
integral  factors. 

83.  Oral 

1.  Give  the  factors  of  51;  45;  99;  87;  96;  69;  84;    91. 

2.  Name  three  factors  of  80. 

3.  Name  as  many  factors  of  24  as  you  can. 

4.  Of  what  numbers  are  3,  5,  and  11  the  prime  factors? 

5.  Name  four  multiples  of  8. 

6.  132  is  the  product  of  11  and  what  other  factor? 

7.  Name  all  the  prime  numbers  smaller  than  132. 

&,    98  is  the  product  of  three  factors.     Two  of  them  are  2 
and  7.     What  is  the  other  ? 

9.    Of  what  number  are  2,  3,  5,  and  7  the  prime  factors  ? 

10.  Give  the  prime  factors  of  35;  45;  81;  63;  57;  38; 
48;  51;  108;  231;  121;  144. 


30  GRAMMAR  SCHOOL   ARITHMETIC 

11.  5,  2,  and  what  other  number  are  the  prime  factors  of  70? 

12.  Give  two  factors  of  30  that  are  not  prime. 

13.  What  even  number  is  prime  ? 

84.  Rule  for  finding  whether  a  Number  is  Prime  or  Composite. 

1.  If  the  given  number  is  odd,  divide  it  hy  3. 

2.  If  Z  gives  a  remainder,  divide  the  given  number  hy  5. 

3.  Continue  this  process,  using  each  prime  number  in  order  as 
a  divisor,  until  an  exact  divisor  is  found,  or  until  the  divisor 
equals  or  exceeds  the  quotient.  If  no  exact  divisor  is  found  until 
the  divisor  used  equals  or  exceeds  the  quotient,  the  number  is  prime. 
Otherwise  it  is  composite.  Even  numbers  need  not  be  tested. 
Why? 

85.  1.  Applying  the  tests  given  on  page  26  instead  of  actu- 
ally dividing  by  3  and  5,  determine  whether  191  is  prime  or 
composite. 

7)191  11)191  13)191  "  17)191 

27-2  rem.  17-4  rem.  14-9  rem.  11-4  rem. 

191  is  not  divisible  by  3  or  5.  (How  do  we  know?)  Since 
the  divisor,  17,  is  greater  than  the  quotient,  11,  and  no  exact 
divisor  has  been  found,  191  must  be  prime. 

Find  whether  each  of  these  numbers  is  prime  or  composite  : 

2.  123  6.    263  10.    197  14.    1618        18.    401 

3.  253  7.    143  11.    217  15.   487  19.    593 

4.  187  8.    721  12.    361  16.    781  20.    3950 

5.  561  9.    407  13,    1005  17.    437  21.   1241 


CANCELLATION  81 

86.    Written 

1.    Find  the  prime  factors  of  7020. 


By  what  kind  of  numbers  do  we  divide  ?    Why  ? 

Which  divisors  do  we  use  first? 

What  besides  the  divisors  is  a  prime  factor? 


2 

7020 

2 

3510 

3 

1755 

3 

585 

3 

195 

5 

65 

13 

.2 

.3.3 

ind 

^  thep 

2. 

112 

3. 

420 

4. 

660 

5. 

nil 

6. 

1055 

7. 

4626 

3  •  5  •  13    Prime  factors,  Ans, 


8. 

145 

14. 

3087 

9. 

129 

15. 

667 

10. 

625 

16. 

310 

11. 

4293 

17. 

399 

12. 

1425 

18. 

1287 

13. 

1414 

19. 

253 

CANCELLATION 

20. 

1682 

21. 

561 

22. 

1001 

23. 

1225 

24. 

6822 

25. 

7290 

87.    Dividing  both  dividend  and  divisor  by  the  same  number 
affects  the  quotient  how  ? 

7 
77 

m 

462     ;2x^x7x^      n  ^    .'    .  M     7  /I    .•    , 

n 
1 

In  either  method  we  divide  both  dividend  and  divisor  by  2,  by  3,  and  by  11 


32  GRAMMAR  SCHOOL   ARITHMETIC 

Taking  out  the  same  factor  from  both  dividend  and  divisor  is 
cancellation. 

88.  Solve  hy  cancellation: 

1.  Divide  36  x  54  x  49  x  38  x  50  by  70  x  18  x  30. 

2.  (28  X  152  X  48)  -^  (14  x  19  x  24  x  2  x  8)  =  ? 

3.  (182  X  5  X  54)  -  (13  X  35  X  6)  =  ? 

4.  What  is  the  quotient  of  108  x  48  x  80  divided  by  27  x 
72  X  40  ? 

5.  Divide  125  x  45  x  7  x  10  by  49  x  5  x  2  x  225. 

6.  Divide  65  x  51  x  11  x  9  x  4  by  17  x  20  x  12  x  11  X  26. 

7.  Divide  25  x  26  x  72  x  14  by  78  x  9  x  120. 

8.  How  many  bushels  of  potatoes  at  80  cents  a  bushel  must 
be  given  in  exchange  for  45  pounds  of  tea  at  64  cents  a  pound? 

9.  (240  X  36  X  385  x  26)  -f-  (12  x  154  x  Qb'). 

10.    What  prime  factor  besides  19  and  11  has  8569? 

LEAST  COMMON  MULTIPLE 

89.  Oral 

1.  3x8  =  ?     24  is  what  of  3?     Of  8  ? 

2.  4  X  6  =  ?     24  is  what  of  4  ?     Of  6  ? 

3.  Name  all  the  numbers  of  which  24  is  a  multiple. 

4.  Define  multiple. 

90.  A  number  that  exactly  contains  two  or  more  numbers  is  a 
common  multiple  of  those  numbers;  e.g.  12  is  a  common  mul- 
tiple of  2,  3,  4,  and  6.  36  also  is  a  common  multiple  of  2,  3, 
4,  and  6. 

Can  you  name  any  other  common  multiple  of  2,  3,  4,  and  6  ? 


LEAST  COMMON  MULTIPLE  33 

91.  The  smallest  number  that  exactly  contains  two  or  more  num- 
bers is  their  least  common  multiple  (L.  C.  M.)  ;  e.g.  18  is  the 
least  common  multiple  of  3,  6,  and  9.  36  is  a  common  multi- 
ple of  3,  6,  and  9.     Why  is  it  not  the  least  common  multiple  ? 

92.  Oral 

Find  the  L,  0.  M.  of: 

1.  5  and  3  10.    5,  4,  and  4 

2.  2,  5,  and  4  li.    7,  4,  and  2 

3.  4  and  10  12.    10,  15,  and  4 

4.  18  and  12  13.    2,  4,  8,  and  12 

5.  20  and  6  14.    12,  5,  and  15 

6.  8  and  12  15.    7  and  12 

7.  5,  6,  and  2  16.    14  and  6 

8.  1,  8,  6,  and  4  17.    2,  15,  6,  and  5 

9.  2,  3,  and  11  18.    4,  18,  3,  and  12 

93.  When  the  least  common  multiple  is  a  large  number,  the 
following  direct  method  is  employed  in  finding  it : 

Let  it  be  required  to  find  the  L.  C.  M.  of  12,  15,  and  18. 

12  =  2x2  x3 
15  =  3x5 
18  =  2x3x3 

What  kind  of  factors  have  we  found?  A  number,  in  order 
to  contain  12,  must  have  what  prime  factors  ?  What  prime 
factors  must  it  have  in  order  to  contain  15  ?   18  ? 

A  number  that  contains  12,  15,  and  18  must  have  how  many 
factors  2  ?     How  many  factors  3  ?     How  many  factors  5  ? 


34  GRAMMAR   SCHOOL   ARITHMETIC 

What  is  the  smallest  number  that  has  the  factors  2,  2,  3,  3, 
and  5  ?     What,  then,  is  the  L.  C.  M.  of  12,  15,  and  18  ? 
The  prime  factors  may  be  easily  found  in  this  way : 


12     15     18 


^5        9  By  what  kind  of  numbers  do  we  divide? 

""5      3        2  X  3  X  2  X  5  X  3  =  180  L.  C.  M. 


94.  Find  the  L.  CM,: 

1.  36,  54,  60  8.  315,  60,  140,  210  15.  70,  15,  30,  14 

2.  18,  24,  36  9.  24,  84,  54,  360  16.  48,  240,  21 

3.  48,  144,  180  10.  75,  20,  35,  120  17.  9,  36,  90,  63,  42 

4.  7,  9,  54  11.  98,  21,  35,  315  18.  25,  15,  60,  50 

5.  72,  40,  48  12.  72,  48,  96,  192  19.  13,  19,  17 

6.  90,  24,  36  13.  120,  18,  20,  60  20.  2,  3,  4,  5,  6,  7,  9 

7.  105,  210,  21,  28  14.  48,  24,  40,  30  21.  21,  56,  45,  70 

GREATEST  COMMON  DIVISOR 

95.  A  number  that  will  exactly  divide  two  or  more  numbers  is 
a  common  divisor  of  those  numbers;  e.g.  5  is  a  common  divisor 
of  30,  40,  and  60. 

96.  The  largest  number  that  will  exactly  divide  two  or  more 
numbers  is  their  greatest  common  divisor  (G.  C.  D.);  e.g.  10  is 
the  greatest  common  divisor  of  30,  40,  and  60. 

Note. — A  common  divisor  is  sometimes  called  a  common  factor,  and  the 
greatest  common  divisor  is  sometimes  called  the  highest  common  factor. 

97.  Numbers  that  have  no  common  divisor  are  prime  to  each 
other;  e.g.  13  and  15. 


GREATEST   COMMON  DIVISOR  35 

98.  Oral 

1.  Findthea.O.D.  of: 

a,  18,  9,  12  e.  m,  24,  40  i.  90,  45,  60 

h,  40,  30,  35  /.  70,  28,  42  /.  54,  27,  36 

c,  14,30,16  g.  33,22,121  k.  60,24,36,48 

d.  36,  30,  18  A.  21,  54,  39  I.  96,  32,  48 

2.  Name  two  numbers  of  which  11  is  a  common  divisor. 

3.  Name  three  numbers  of  which  12  is  a  common  divisor. 

4.  Name  two  numbers  which  are  prime  to  each  other. 

5.  What  is  the  greatest  number  that  will  exactly  divide  84, 
60,  and  36  ? 

6.  Name  two  numbers  of  which  13  is  the  G.  C.  D. 

7.  Tell  which  of  these  pairs  of  numbers  are  prime  to  each 
other : 

a.    12  and  49     h.    48  and  60       c.    38  and  63     d.    16  and  45 

99.  Written 

1.    Find  the  greatest  common  divisor  of  336,  504,  and  924. 

336  =  ;2x;2x2x2x^x  7 
504  =  ;2x?x2  x^x3x7 
924  =  ;2  X  ;2  X  ^         X  7  X  11 

2  x2x3  x7  =  84G.C.D. 

Factoring  the  numbers  and  selecting  the  common  prime  factors,  we 
find  them  to  be  2,  2,  3,  and  7.  Since  all  of  them  are  factors  of  each  of 
the  given  numbers,  their  product,  84,  is  the  greatest  common  divisor 
required. 


I 


36  GRAMMAR  SCHOOL   ARITHMETIC 

The  common  prime  factors  may  easily  be  found  in  this  way : 


2 

336 

504 

924 

2 

168 

252 

462 

3 

84 

126 

231 

7 

28 

42 

77 

4 

6 

11 

2  •  2  •  3  •  7  Common  prime  factors. 
Find  the  a.  C,  B. : 

2.  84,  126                 8.  252,  96,  120,  24  14.  378,  126,  189 

3.  180,  210               9.  120,  168,  216  15.  144,  243,  135 

4.  448,  168             10.  90,  270,  160  16.  364,  143,  312 

5.  396,  468             11.  305,  60,  90  17.  576,  400,  240 

6.  280,  60,  80         12.  180,  72,  81  18.  168,  630,  616,  350 

7.  320,  144,  560     13.  176,  121,  165  19.  1980,  945,  245 

20.  Find  the  greatest  number  that  will  exactly  divide  567, 
378,  and  504. 

21.  Find  all  the  common  prime  factors  of  630,  720,  and  540. 

22.  Find  the  product  of  all  the  common  prime  factors  of  216, 
432,  and  720. 

23.  Find  a  number  that  is  prime  to  350. 

24.  Name  three  numbers  of  which  13  is  the  greatest  common 
divisor. 

FRACTIONS 

100.  One  or  more  of  the  equal  parts  of  a  unit  is  a  fraction  ; 

p  ri    1  •    3  .    2  .    at  5 
^'9'    8'     5'    T'    '^TO- 

101.  A  fraction  is  always  an  expression  q/ division.     For  ex- 
ample, if  1  inch  is  divided  into  8  equal  parts,  each  part  is  J  of 


FRACTIONS  .  37 

an  inch.  If  a  line  7  inches  long  is  divided  into  8  equal  parts, 
one  part  is  |  of  an  inch  long.  That  is,  1  inch  ^  8  =  |  inch, 
and  7  inches  -?-  8  =  |^  inch. 

Take  your  rule  and  draw  a  line  1  inch  long.  Divide  it  into 
4  equal  parts.  How  long  is  one  part  ?  Draw  a  line  3  inches 
long.  Divide  it  into  4  equal  parts.  Measure  one  of  the  parts. 
3  inches  -j-  4  =  ? 

Draw  a  line  5  inches  long.  Divide  it  into  8  equal  parts. 
Measure  one  of  the  parts.  5  inches -^8  =  ?  3-j-7  =  ? 
9-4-11  =  ? 

102.  The  number  above  the  line  in  a  fraction  is  the  numerator. 
It  is  always  a  dividend.  In  the  fractions  ^,  ^,  ^^-,  -H,  the 
numerators  are  1,  7,  15,  and  23. 

103.  The  number  below  the  line  in  a  fraction  is  the  denominator. 
It  is  always  a  divisor.  In  the  fractions  ^,  |^,  ^g^-,  ^|,  the  de- 
nominators are  3,  9,  5,  and  12. 

104.  The  numerator  and  the  denominator  are  the  terms  of  a 
fraction  ;  e.g.  the  terms  of  -^^  are  7  and  11. 

105.  The  value  of  a  fraction  is  the  quotient  obtained  by  divid- 
ing the  numerator  by  the  denominator. 

REDUCTION  OF  FRACTIONS 

106.  Changing  the  form  of  a  number  without  changing  its  value 
is  reduction  ;  e.g.  8  pt.  =  4  qt. ;    i7  =  700  ct. ;    7  ft.  =  ^  yd. ; 


REDUCTION   TO  LOWEST  TERMS 

107.    A  fraction  is  in  lowest  terms  when  the  numerator  and  de- 
nominator are  prime  to  each  other ;  e.g.  -f^,  ||,  ^^. 


38  GRAMMAR  SCHOOL  ARITHMETIC 

108.    Oral 

1.  A  fraction  is  always  an  expression  of  what  operation  ? 

2.  Tlie  numerator  of  a  fraction  is  which  term  in  division  ? 
The  denominator  ?     The  value  of  the  fraction  ? 

3.  Dividing  the  dividend  and  the  divisor  by  the  same  num- 
ber affects  the  quotient  how  ? 

4.  Dividing  the  numerator  and  the  denominator  of  a  fraction 
by  the  same  number  affects  the  value  of  the  fraction  how  ? 


Summary 

A  fraction  may  he  reduced  to  lowest  terms  hy  dividing  its 
terms  hy  their  common  factors,  continuing  the  process  until  the 
terms  are  prime  to  each  other ;  or,  hy  dividing  hoth  terms  of  the 
fraction  hy  their  greatest  common  divisor. 

109.     Written 

1.    Reduce  1||  to  lowest  terms. 

163  —  3 3  ~  A* 

We  divide  both  terms  by  5  and  then  by  3.  If  the  greatest  common 
divisor,  15,  is  used,  only  one  division  is  necessary. 


Reduce  to  lowest  terms  : 

2-  \n 

6-     2WU 

10. 

tV\% 

14. 

A¥r 

18. 

Ml 

3-  m 

'•     lV5% 

11. 

iifl 

15. 

li 

19. 

iW 

*•  m 

8-    fsV* 

12. 

\%\ 

16. 

if 

20. 

m 

5-    iff 

9-    i-h% 

13. 

ill! 

17. 

i! 

21. 

m 

FRACTIONS  39 

22.  Express  in  lowest  terms  637  -?-  833. 

23.  Express  in  lowest  terms  the  quotient  of  288  divided  by 
504. 

24.  Express  in  lowest  terms  f^f  • 


REDUCTION    OF    IMPROPER    FRACTIONS    TO    INTEGERS    OR 
MIXED  NUMBERS 

110.  A  fraction  whose  numerator  is  smaller  than  its  denomi- 
nator is  a  proper  fraction ;  e.g.  |,  ^l^,  -^f .  The  value  of  a  proper 
fraction  is  always  less  than  1. 

111.  A  fraction  whose  numerator  equals  or  exceeds  its  denomi- 
nator is  an  improper  fraction,  e.g.  f ,  |,  ^^.  The  value  of  an 
improper  fraction  compares  how  with  1  ? 

112.  A  number  that  is  composed  of  an  integer  and  a  fraction  is 
a  mixed  number;  e.g.  5f,  lOJ,  201^6-,  18.25. 

113.  Oral 

1.  A  fraction  is  an  expression  of  what  operation? 

2.  Define  the  value  of  a  fraction. 

3.  The  value  of  an  improper  fraction  is  always  an  iflteger  or 
a  mixed  number.     How  may  we  find  it? 


Find  the  values  of: 

4.      1 

8.    \^ 

12. 

¥ 

16. 

!! 

20. 

!l 

24. 

fl 

5-      f 

9.    J/ 

13. 

¥ 

17. 

¥ 

21. 

\\ 

25. 

\i 

6.     1 

10.    Y 

14. 

¥ 

18. 

1|A 

22. 

!l 

26. 

w 

7.   Y 

11.    ^- 

15. 

¥ 

19. 

i3A 

23. 

!l 

27. 

w 

I 


) 

GRAMMAR  SCHOOL   ARITHMETIC 

114. 

Written 

1. 

H^ 

3.     ^^^ 

5. 

¥2^ 

7.    ^IF 

9. 

W 

2. 

¥# 

^-  ',V 

6. 

^¥- 

8.    ^iV- 

10. 

\W 

11. 

■¥/ 

14.  m^ 

17. 

¥/- 

20. 

^IF 

12. 

mi 

15.    w 

18. 

Y/ 

21. 

X|-|l 

13. 

w 

16.     ^^ 

19. 

\W 

22. 

Hli' 

REDUCTION   OF  INTEGERS  AND  MIXED  NUMBERS  TO 

IMPROPER  FRACTIONS 
115.     Written 
1.    Reduce  38|-  to  a  fraction. 

38  =  38  X  9  ninths  =  342  ninths. 
342  ninths  plus  7  ninths  =  349  ninths. 
The  work  may  be  expressed  thus :     38|  =  ^^  Ans, 

9 


342 

7' 
349 

Reduce  to 

improper  fractions  : 

2.   17f 

9.    26^3^ 

16. 

1253V 

23. 

217,V 

3.    15| 

10.   45if 

17. 

159f 

24. 

248f 

4.   29if 

11.   571 

18. 

167f 

25. 

459^ 

5.    25^^ 

12.    255 

19. 

24M 

26. 

160,V\ 

6.   69| 

13.    35^\ 

20. 

55|i 

27. 

388t^ 

7.   170f 

14.    57f, 

21. 

129^V 

28. 

646fA 

8.   49| 

15.    61| 

22. 

216U 

29. 

569|J 

FRACTIONS  41 

30.  In  560  there  are  how  many  5ths? 

31.  Reduce  250  to  16ths. 

32.  Change  12|  to  16ths. 

33.  Change  156  to  a  fraction  whose  denominator  shall  be  12. 

LEAST   COMMON   DENOMINATOR 

116.  Fractions  whose  denominators  are  alike  have  a  common 
denominator ;  e.g,  60  is  a  common  denominator  of  -f-^^  l|^,  and  |^. 

117.  Fractions  having  the  smallest  possible  common  denominator 
have  their  least  common  denominator ;  e.g.  ^^,  ^q,  gV 

118.  Oral 

1.  We  have  found  that  when  we  add  fractions  having  dif- 
ferent denominators,  we  must  first  change  them  to  fractions 
having  the  same  denominator.  What  shall  we  call  that 
denominator? 

2.  Since  the  common  denominator  must  contain  all  the  given 
denominators,  it  must  be  what  of  those  denominators?  (A 
number  that  exactly  contains  two  or  more  other  numbers  is 
what  ?) 

3.  The  least  common  denominator^  then,  must  be  which  mul- 
tiple of  the  given  denominators? 

4.  Reduce  |,  |,  and  f  to  fractions  having  the  least  common 
denominator. 

119.  Written 

Change  j^,  j^^,  ^J,  and  ^^  to  fractions  having  the  least  com- 
mon  denominator. 


i 


42 


GRAMMAR  SCHOOL   ARITHMETIC 


7 

8 

16 

17 

2 

10 

15 

33 

30 

3 

5 

15 

33 

15 

5 

5 

5 

11 

5 

1   11 


330-5-10  =  33 


330  -H  15  =  22 


330-^33  =  10 


2x3x5xll  =  330L.  C.  M.  330-30  =  11 


7  X  33  ^  231 
330 

^176 

15  X  22     330 
16x10^160 
330 
187 


10x33 
8  x22 


33x10 
17x11 


30  X  11      330 


2  31   115.     160     18  7     A^fi 

3  3  0'  3  3  0'    3  3  0'   33  0  ^'''*' 

Change  to  fractions  having  the  least  common  denominator: 


sf  I 


2.    |,l,i 


3-  hx%i 
5-  ftVM'A 


6-    if  f-l 
»•     2'  T'  6'   12 


*. 


4     __9_ 
9'    10 


9. 

10-  i'l'A'f'l 


11  14.    15     .3  7 

"^■*-  3  5'    91'   65 

•*-2-  tV  I'  h  i J 

13  j5     1A    _5_      3 

■^'''  6'   34'    12'   lY 

14. 


4     21      7      _5 
9'  26'   l¥'   1 


15       2     __8_     _5_    JL    __8_ 
■^^-     Y'  21'   13'   63'    11 


ADDITION   OF   FRACTIONS   AND    MIXED    NUMBERS 

120.  A  number  is  in  its  simplest  form  when  it  is  in  the  form  of 
an  integer^  or  a  proper  fraction  in  its  lowest  terms,  or  a  mixed 
number  whose  fractional  part  is  in  its  lowest  terms ;  e.g.  18,  |, 
and  51  are  in  simplest  form  ;  -2^-,  |i  ^^,  and  8|  are  not.     Why  ? 

Answers  should  always  be  expressed  in  simplest  form, 

121.  Written 


1.    Add  i  ^9^,  and  IJ. 


2 
2 

3 

'34       1 
2x2x3x3x4 


9 

16 

12 

9 

8 

6 

9 

4 

3 

1—1 


HI 


144,  L.  C.  D. 


^l^  =  2^iSum 


FRACTIONS  43 

2.    Add  lOf,  7f ,  and  6f . 

75  ~~    72  5  *  ^®    ^^^    *^®    integers    and    frac- 

*■§■  ~     ^40  tions  separately,  and   then  unite  the 

6f=    6|^  sums. 

Rule.  —  To  add  fractions,  reduce  them  to  their  least  common 
denominator^  add  the  numerators,  place  the  sum  over  the  common 
denominator,  and  reduce  the  result  to  simplest  form. 

When  there  are  integers  or  mixed  numbers,  add  the  integers  and 
the  fractions  separately,  and  unite  the  results. 

Add: 

5-   h\h\hU>ii  10-    X2f,19f,28^ 

6.  J,|,^,H  11.    19f,  18|,  15J,  12Jj 

7.  16J^,  24J3,  43ii  12.    71,9,75,61,41 

13.  During  a  storm,  a  tree  was  broken  off  17f  feet  from  the 
ground.  The  piece  broken  off  was  41|  feet  long.  How  tall 
was  the  tree  before  it  was  broken  ? 

14.  The  subtrahend  is  89||-  and  the  remainder  49^^.  What 
can  you  find  ?     Find  it. 

15.  A  rectangular  field  is  509^^  feet  long  and  347^g  feet 
wide.     How  many  feet  of  fence  are  re(}uired  to  inclose  it? 

SUBTRACTION  OF  FRACTIONAL  AND  MIXED  NUMBERS 

122.  Fractions  must  have  a  coi^Qaoii'- cfeii'orainator  in  order 
that  one  may  be  subtracted  from  tMe  ot^reK 

i 


i 


44  GRAMMAR  SCHOOL   ARITHMETIC 

123.     Written 

1.    From  il  take  |. 

11  =  11 


1  ^       t  J  How  is  45  obtained  ? 

5 

il  Difference 


4  =  |4 


In  subtracting  mixed  numbers,  if  the  fraction  in  the  subtrahend  is 
greater  than  that  in  the  minuend,  one  integral  unit  of  the  minuend  must 
be  united  with  the  fraction  to  form  an  improper  fraction,  before  subtracting. 

2.    From  291  take  IS^^- 

291  =  29^0  =  28f f  How  do  we  obtain  |f  ? 

15|^|  =  15^  Differ enee 

3.  121 -f  11.  11  _  I  19.  IS^V-l^^V 

4.  261  -  4f  12.  181  -  151  20.  381^3^  -  332^5^ 

5.  242-1  -  33f  13.  1101  -  109|  21.  62|  -  46-f- 

6.  298f  -  149|  14.  1121  -  151  22.  3301  _  140_3- 

7.  43|  -  22|  15.  17|  -  15f  23.  189|  -  1431 

8.  26f  -  8|  16.  146f  -  127|  24.  407^^  -  3981 

9.  f  -  I  17.  167f  -  76f  25.  90|  -  484 
10.  1|  -  1^  18.  421  -  36f  26.  81^4^  -  371 

MULTIPLICATION  AND  DIVISION  COMBINED 
124.      Written 

(20  -5-  4)  X  (21  -  7)  =  ? 


15     Ans, 


tXM  aHA  Cor 


4       7        4^7iteW»'- 


\ 


FRACTIONS  45 

20  and  21  are  dividends  and  4  and  7  are  divisors.  The  result  is  the  same 
whether  we  make  each  division  separately  and  then  multiply  the  quotients, 
or  divide  the  product  of  the  dividends  by  the  product  of  the  divisors.  In 
many  cases  the  latter  way  is  easier,  because  we  may  use  cancellation ;  e.g. 

..  (20.4)  X  (21.7)  =  (fxf)=M  =  15^..; 

^       7       42 
h.  (18  .  7)  X  (28  .  24)  X  (210  -  15)  =  ^l^f.^  ^^^  =  42  Ans. 

Find  results : 

1.  (22  -  11)  X  (12  -f-  5)  X  (25  .  6)  x  (25  .  2) 

2.  (16  .  4)  X  (20  .  6)  X  i^S  .  10)  X  (42  -  11) 

3.  (52  .  13)  X  (35  -V-  21)  X  (12  -  7)  x  (21 .  3) 
^     28     42     36     63 

^-  y^T^y^ii 

5.  (36  .  27)  X  (35  .  75)  x  (25  .  12)  x  (12  -  7) 

6.  (7  .  49)  X  (68  .  7)  X  (14  .  8)  x  (35  .  17) 

7.  (40  ^  39)  X  (52  -f- 10)  X  (34  .  13)  x  (125  .  10) 

8.  (70  .  35)  X  (26  .  20)  x  (68  .  13)  x  (125  .  35) 

9.  (70  .  17)  X  (68  .  24)  x  (35  .  7) 

_    49     75      108      98         „     51     49      24     17      20 
^"-    25^r2^^^15         ''•    60V6^34^-5-^¥ 

12.  Multiply  the  quotient  of  79  divided  by  24  by  the  quo- 
tient of  168  divided  by  79. 

MULTIPLICATION  OF   FRACTIONS 

125.  Any  integer  may  he  expressed  as  a  fraction  hy  writing 
it  as  a  numerator  with  \  for  a  denominator ;  e.g.  5  is  the  same  as 
f  ;  19  is  the  same  as  ^-^-\  |  X  7  x  |f  is  the  same  as  f  X  ^  x  ||. 


46  GRAMMAR  SCHOOL  ARITHMETIC 

126.  The  word  of,   between  fractions^  means    the  same  as  the 
sign  of  multiplication ;    e.g.  |of  |  =  |x|;  ^oi4iX^Q  =  ^x\ 

■^   16'     2     •     3   ^^   5  2     •    V3    -^    5/' 

127.  An  indicated  multiplication  of  two  or  more  fractions  is 
called  a  compound  fraction ;  e.g.  f  X  f ;  le  '^  M  ^  11 '  Y  ^^  f  * 

128.  Written 

1.  Find  the  product  of  |,  f ,  and  -^^, 

Each  of  these  fractions  indicates  what  operation  ? 

Since  all  the  numerators  are  dividends  and  all  the  denominators  are  divi- 
sors, we  may  find  the  result  by  dividing  the  product  of  the  numerators  by 
the  product  of  the  denominators,  using  cancellation : 

3 
2     5      9      15    . 

8 
Find  the  products : 

2.  |X|  8.    f  of  ^9^  off        14.    AxfX^xil 

3.  ^Vxf  9.    JxJjXf  15.    foffoffxl4 

4.  I  of  I  of  ^\  10.  T^  X  I  X  I  16.   I  X  t:\  X  if  X  22 

5.  Joffofif  11.  Mx-V^xf  17.    ix2x|ofJj 
6-  ix^VxA  12.  if  x34xf  18.  f^XT^of84xT^e 
7.  f  of  J  of  I  13.  f  off  of  15  19.   ifx/jx^ 

129.  Mixed  numbers  may  be  reduced  to  improper  fractions  and 
then  multiplied;  thus, 

l|x8iXiVx4  = 
2 
^x^'^x   ^   x^-^^-162  ^/IS 


FRACTIONS 


47 


6.    Find  ^6_  of  f  of -11  of  8| 


7.    8ix8f 


3.  77fx3 

4.  85fx47f 


Written 

1.  6fx4| 

2.  12^\x7J 

8.  15|x8fx^\ 

9.  5^  X  5f  X  f 

5.    781  X  17|  10.   -^x^x^x  6f 

11.  Multiply  I  by  101  by  f  by  f  by  6|. 

12.  Multiply  :  a,    25|  by  24f .     5.    116  by  |. 

13.  15|  X  124  X  20       15.    9|  X  ^V  X  21       17.   |  x  4  x  5i 

14.  171  X  15f  X  f         16.    61  X  11  X  i\       18.   -^x^Ox  5| 

130.  In  multiplying  a  large  mixed  number  by  an  integer, 
time  may  be  saved  by  multiplying  the  whole  number  and  the 
fraction  separately,  then  adding  the  products,  thus  : 

Written 

1.    845^^x8  =  ?  r 


845A 

8         ,5^x8  =  3,:^ 
3  j^      845  X  8  =  6760 

6760          6760  +  3,^  =  6763tV  Ans. 

6763^ 

9. 

381^5^  X  27 

2.    89|x5 

10. 

3079f  X  15 

3.    20811x6 

11. 

413^^  X  20 

4.    628fxl5 

12. 

6283|  X  18 

5.    830^\xl8 

13. 

3100^5  X  35 

6.    2037-jVx28 

14. 

2050^  X  52 

7.    3547fxl00 

15. 

83101  X  51 

8.    230J^x200 

16. 

2806i|  X  90 

48  GRAMMAR  SCHOOL   ARITHMETIC 

DIVISION  OF  FRACTIONS 

131.  Divide  f  |  by  f . 

Since  ||  is  a  product  and  |  is  one  of  its  factors,  we  may  state 
the  question  thus :      4 

35^5     o     ^  35  _  5  X  ? 

72     8      ■  ^"^  72     8  X  ? 

In  order  to  find  the  required  factor  we  must  divide  the 
numerator  35  by  5,  and  the  denominator  72  by  8,  thus : 

35^^7 

72 -V- 8     9* 

That  is  exactly  what  we  should  do  if  the  question  were : 

7 

72     5      '        Jf     §     9' 
9 

The  latter  method  is  the  more  convenient,  especially  when 
the  numerator  of  the  divisor  is  not  exactly  contained  in  the 
numerator  of  the  dividend,  or  the  denominator  of  the  divisor  in 
the  denominator  of  the  dividend. 

Therefore,  to  divide  by  a  fraction  we  interchange  the  terms  of 
the  divisor  and  multiply, 

132.  Written 

1.    Divide  4|  by  5|. 

Solution:  4f  ^ 5f  =  M^|=  M  x  A  =  |  Ans. 

2 
How  do  we  treat  mixed  numbers  before  dividing? 


FRACTIONS 


49 


2.    Divide  47  by  6|. 

Solution :  47  -^  6^  =  ^^^  --  -i  3  =  4JL 


KJUVf^VVUIV  ,       1  1     - 

-  1^2 

—  ~1    ~~2~ 

~     1      ' 

"^  13  =  It  = 

'T3 

j±ns. 

How  do  we  treat 

integers  ? 

3-    t\^A 

9. 

3J^if 

15. 

18-^,V 

21. 

n^\i 

4.    f^l 

10. 

6i^A 

16. 

15-1 

22. 

4i^3J 

S-   T^  +  f 

11. 

2i\^5| 

17. 

1^14 

23. 

7i  +  6f 

6-    I's^f 

12. 

^  +  4i 

18. 

if^8 

24. 

If  +  Jf 

7-    If^l 

13. 

i|-^6| 

19. 

21^51 

25. 

fi^T^ 

8-   it-A 

14. 

2  +  1 

20. 

71-lJ 

26. 

121^41^ 

27.  By  what  must  ||  be  multiplied  to  make  ||^? 

28.  One  factor  of  f  A  is  li      What  is  the  other? 

29.  a.    Jl  X  ? 


9_         J       ?   X    2  2   _  __4_ 
2-       ^-      •    ^   89  —   15* 


30     a     ^i  =  li^  X  *?     5     -4JL 

ow.     c*.      gg  —  J^36    '^    •       ^'209 


tix? 


133.  A  fraction  whose  terms  are  integers  is  a  simple  fraction ; 
e.g.  \^  is  a  simple  fraction. 

134.  A  fraction  that  has  a  fraction  in  either  or  both  of  its  terms 

3      2.     5 1    g2.  ^  ^  9 

is  a  complex  fraction;  e.g.  — -,  -^,  -f ,  -f ,  and  .^^  '  ^  are  com- 

8f    16   25    7|  lf-| 

plex  fractions.     Read  each  fraction. 

A  complex  fraction  is  merely  an  indicated  division  of  frac- 
tions, made  by  writing  the  dividend  above  a  line  and  the  divi- 
sor below  the  line,  just  as  a  simple  fraction  is  an  indicated 
division  of  integers ;  therefore, 

A  complex  fraction  mag  he  reduced  to  a  simple  fraction  by 
dividing  the  expression  above  the  line  by  the  expression  below  the 
line. 


50  GRAMMAR   SCHOOL   ARITHMETIC 

135.     Written 

1.  Reduce  —  to  a  simple  fraction. 

2.  Reduce  32  to  a  simple  fraction. 

40 

&=A^40  =  i-x— =  —  Ans, 
40      17  11     iP     136 

8 

74 

3.  Reduce  —f~  to  its  simplest  form. 

m 

5 

2-1!"        '     ''"  8    •  20"  ^  ''5a~106~^^^6^'''* 

2 

iw  examples  4-13  change  the  given  complex  fractions  to  simplest 
form  : 

4   li       6   M      8  il     10  ^i-     12   i^ 

*•    if  ^-      6  °-    12t       '°-    |xl2i       12.    j^^.^ 

8i  i|  4  24h-4  I  of  Si- 

14.   If  I  of  an  acre  of  land  is  worth  f  72-^^^,  what  is  the  value 
of  3  acres  at  the  same  rate  ? 

15.  There  are  5J  yards  in  a  rod.     How  many  rods  in  140| 
yards  ? 

16.  At  |6|  a  ton  how  many  tons  of  coal  can  be  bought  for 

i77f? 


DECIMALS   AND  COMMON   FRACTIONS  51 

136.  In  division,  if  the  divisor  contains  a  common  fraction 
that  cannot  easily  be  reduced  to  a  decimal,  it  is  sometimes 
helpful  to  multiply  both  dividend  and  divisor  by  the  denomi- 
nator of  the  fraction,  thus  making  both  dividend  and  divisor 
integers,  or  simple  decimals;  e.g.: 


.021i).416 

Multiplying  both  dividend  and  divisor  by  3,  and  then 
dividing, 

19.5  Quotient 

.064)1.248-0 
Written 

137.  In  the  following  examples  find  the  quotients  correct  to  two 
decimal  places  : 

1.  8.48  ^19|  5.  28.9 -^Tf  9.  T.9f^4^ 

2.  3.56-V-41I  6.  30.05 -5- .17f  lo.  9.375 -^.16f 

3.  9.305 ^9f  7.  8.3--.07f  ii.  3.23-^1.21 

4.  35.3125 -^12f  8.  .0135-^.021^  12.  .484-^.5^ 

COMPARATIVE  STUDY  OF  DECIMALS  AND  COMMON  FRACTIONS 

138.  A  fraction  that  is  expressed  hy  writing  the  numerator 
above  and  the  denominator  below  a  line  is  a  common  fraction; 
e.g.  f ,  ||.      (See  §  9  for  definition  of  decimal  fractions.) 

All  decimal  fractions  may  be  expressed  as  common  fractions 
without  reducing  them;  e.g.  .0104  =  -jl^fo.  What  common 
fractions  can  be  expressed  as  decimals  without  reducing  them  ? 

139.  When  a  decimal  fraction  is  expressed  without  its  de- 
nominator, by  using  the  decimal  pointy  it  is  said  to  be  expressed  in 
the  decimal  form. 


52  GRAMMAR  SCHOOL   ARITHMETIC 

Oral 

.7      =  J7_^  or  7  divided  by  10 

.305  =  ^3JL5_,  or  305  divided  by  1000 
.581  =  ^,  or  581  divided  by  100 
In  like  manner  tell  the  meanings  of  the  following  decimals: 


1. 

.18 

6. 

.1891 

11. 

.29.1 

16. 

.0051- 

2. 

.41 

7. 

.161 

12. 

.007,2, 

17. 

.0034 

3. 

.216 

8. 

.2391 

13. 

.03f 

18. 

.165 

4. 

.879 

9. 

.548,6, 

14. 

.51341 

19. 

.00017J 

5. 

.200 

10. 

.731 

15. 

.40701 

20. 

.OOOf 

140.  A  decimal  may  he  reduced  to  a  common  fraction  in  sim- 
plest form  hy  expressing  it  as  a  common  fraction  and  reducing 
to     lowest    terms:     e.g.     .  85  =  ,8_5_  =  ii ;      13.8  =  13,%  =  13| ; 

*     ^     100       3       l^^     6 

2 

141.  Written.  Reduce  the  following  decimals  to  common  frac- 
tions or  mixed  numbers  in  simplest  form  : 


1. 

.28 

9. 

.875 

17, 

.0054 

25. 

.003| 

2. 

.125 

10. 

.375 

18. 

.250 

26. 

.1251 

3. 

.235 

11. 

.55 

19. 

.1375 

27. 

.871 

4. 

.75 

12. 

.0025 

20. 

.04f 

28. 

.66f 

5. 

.164 

13. 

.56 

21. 

.121 

29. 

.1361 

6. 

.82 

14. 

.68 

22. 

.621 

30. 

116.25 

7. 

.138 

15. 

16.075 

23. 

.061 

31. 

2.33^ 

8. 

.425 

16. 

.0125 

24. 

.018^ 

32. 

.031 

DECIMALS  AND 

COMMON   FRACTIONS 

53 

33. 

22.621 

38.     .07i\ 

43.     .1621 

47. 

179.00| 

34. 

7.0871 

39.     .126| 

44.     40.40f 

48. 

S.OOff 

35. 

6.131 

40.     .12f 

45.     61.411 

49. 

890.901 

36. 

58.061 

41.    .166| 

46.    42.  If 

50. 

8.000|^ 

37. 

49.6| 

42.     .19,7, 

142.  Since  a  fraction  is  an  expression  of  division,  a  common 
fraction  may  he  reduced  to  a  decimal  by  dividing  its  numerator  by 
its  denominator. 

Before  dividing,  place  a  decimal  point  after  the  dividend. 
Annex  ciphers  as  they  are  needed  ;  e.g. 

,^  =  7. 0000  ^16  =  .4375 
39^  =  39.4375 

143.  Written.     Reduce  to  decimals? 


1. 

1 

11. 

3tV 

21. 

^ 

31. 

19iVk 

2. 

1 

12. 

A 

22. 

iVi 

32. 

625 

3. 

i 

13. 

2% 

23. 

z^ 

33. 

-,h 

4. 

1 

14. 

h 

24. 

2A 

34. 

12ef5 

5. 

1 

8 

15. 

M 

25. 

132^ 

35. 

14M 

6. 

1 

16. 

II 

26. 

19A 

36. 

9iAt> 

7. 

1 

17. 

M 

27. 

12A 

37. 

13A 

8. 

1^6 

18. 

«     , 

28. 

600 

38. 

SAV 

9. 

a 

19. 

160 

29. 

1^ 

39. 

5A 

10. 

h 

20. 

^ 

30. 

t¥b 

40. 

sMo 

144.  A  fraction  in  lowest  terms  whose  denominator  contains 
other  prime  factors  than  2  and  5  cannot  be  reduced  to  an  exact 
entire  decimal;  e.g.  |,  |,  J|,  ,\,  Jf,  ||. 


54  GRAMMAR  SCHOOL   ARITHMETIC 

Such  a  fraction  may  be  reduced  to  a  decimal  of  nearly  the 
same  value  by  carrying  the  division  to  a  certain  number  of 
decimal  places,  thus  : 

Reduce  J|  to  a  decimal  of  four  places. 

.7307^9^  Ans.       .7307  is  almost  equal  to  |f . 
26.)19.0000  The  exact  value  of  i|  is  .7307^93. 

The  result  may  be  expressed,  .7307  +  . 

Written.  Reduce  to  decimals  of  three  places : 
2.    f  8.    A 

4.    I  10.    ^^ 

6.    If  12.    5| 

A  COMMON   FRACTION  AT   THE   END  OF  A   DECIMAL 

145.    .2i  =  .2+(lof  J^,  or^V,or.05). 
.2 +  .05  =.25. 

In  a  similar  manner  we  may  show  that, 

.  27|  =  .  275,         .  384 1  =  .  3845,  etc. 

Also,  that  .  21  =  .225,  .  34 1  =  .  3425,  etc. 

Also,  that  .8|  =  .875,  .06|  =  .0675,  etc. 

Also,  that  .9 J  =  .9125,         .07f  =  .07375,  etc. 

Oral.     Express  as  entire  decimals : 

1.  a.    .8|         b.    1.471       c.    .5601  ^.  27^  e.  .04^ 

2.  a.    .9^         b.   S.S^         c.    $9,001  d.  1.039^  e.  .0145| 

3.  a.    .02f       b.    21. If       c.    $21.06|  d.  .0033f  e.  .0090| 


13. 

8l\ 

19. 

62M 

14. 

331 

20. 

Iff 

15. 

68V_ 

21. 

IjVV 

16. 

53f, 

22. 

W3A 

17. 

Jl 

23. 

216/t 

18. 

A 

24. 

Mf 

ALIQUOT   PARTS  55 

4.  a.    All       ^'    ^'%         ^-    80.03-1-        d.    212f         e.    61. 9^ 

5.  a.    6421      5.    63.97J      c.    15.0f  (^.    24.00-|-      e.    29.00J 

6.  a.    1.40|    5.    2.25f       c.    lO.OJ  d.    25f  e.   4.0001 

ALIQUOT  PARTS 

146.  One  of  the  equal  parts  of  a  number  is  an  aliquot  part  of 
that  number;  e.g.  8  oz.  is  an  aliquot  part  of  16  oz.  because  8  oz. 
is  J  of  16  oz.;  16-|  cents  is  an  aliquot  part  of  100  cents  because 
16|  cents  =  J  of  100  cents. 

Find  the  number  of  cents  in  |1;  |1;  || ;  |i ;  f  1;  |i ;  $1 ; 

iyO'    ^tV'    ^2V- 

The  answers  you  have  given  are  all  what  kind  of  parts  of  a 
dollar  ? 

Prove  the  correctness  of  the  following  table  : 

PARTS   OF   A   DOLLAR 

5    cents  =  I2V  SSJ  cents  =  $J 

6|  cents  =  iy^g  37^  cents  =  8f 

8i  cents  =  f  ^3  ^^    cents  =  IJ 

10    cents  =  I Jq  621  cents  =  f f 

121  cents  =  ^  66|  cents  =  If 

16f  cents  =  |l  75    cents  =  S| 

25    cents  =  f  ^  871  cents  =  $  J 

This  table  should  be  committed  to  memory  like  the  multiplication  table, 
because  its  use  will  shorten  many  problems  ;  e.g.  33  books,  at  $.16f,  each, 
will  cost  33  X  $^  =  $5|. 

When  handkerchiefs  are  \2\^  apiece,  $3  will  buy  as  many  handkerchiefs 

as  $3  ^  .| ^,  or  $ 3  x  f  =  24  handkerchiefs.     Ans. 


56  GRAMMAR  SCHOOL  ARITHMETIC 

147.  Oral 

1.  i.l4|^  =  what  part  of  a  dollar? 

2.  1^  of  a  dollar  are  how  many  cents ?     |?     |-?     |-?     |-? 

3.  20  cents  are  what  part  of  a  dollar  ?    40  cents  ?    60  cents  ? 
80  cents  ?     Which  of  these  is  an  aliquot  part  of  f  1  ? 

4.  Mention  three  aliquot  parts  of  12  ;  two  aliquot  parts  of 
10  ;  five  aliquot  parts  of  64. 

5.  Give  four  numbers  of  which  8J  is  an  aliquot  part. 

6.  What  is  the  cost  of  28  pineapples  when  they  are  bought 
at  the  rate  of  ^.14|^  apiece  ? 

7.  At  $ .  331  a  pound  how  many  pounds  of  butter  will  $  5  buy  ? 

8.  A  man  bought  five  dozen  cans  of  corn  at  the  rate  of  8J 
cents  apiece.     What  did  they  cost  ? 

148.  Written 

1.  Find  the  cost  of  the  following : 

a.  166  pounds  of  pork  at  12|  cents  a  pound. 

b.  248  lb.  of  veal  at  16|  cents  a  pound. 

c.  148  boxes  of  strawberries  at  25  cents  a  box. 

d.  250  lb.  of  butter  at  37|  cents  a  pound. 
.    e.  150  lb.  of  honey  at  25  cents  a  pound. 

/.  640  bars  of  soap  at  6^  cents  a  bar. 

g.  960  dozen  of  eggs  at  $.16|  a  dozen. 

h.  32  yd.  of  dress  goods  at  1.33^  a  yard. 

{.  328  grammar  school  arithmetics  at  $ .  62^  apiece. 

y.  656  steel  shovels  at  1. 87|^  each. 

2.  At  $ .  33 J  a  yard,  how  many  yards  of  linen  can  be  bought 
for  1150? 

3.  How  many  bushels  of  barley  can  be  bought  for  |624,  at 
$.75  a  bushel? 


SPECIAL   CASES  IN  MULTIPLICATION  57 

4.  At  |.66|  each,  how  many  pocket  knives  can  be  purchased 
for  164? 

5.  When  butter  is  25  cents  a  pound,  how  many  pounds  can 
bebought  for  1650? 

6.  How  many  articles,  at  14|  cents  each,  can  be  purchased 
for  1154? 

7.  At  87 J  cents  each,  how  many  books  can  be  bought  for 
fl456? 

8.  How  many  boxes  of  berries  can  be  bought  for  $250,  at 
16|  cents  a  box? 

9.  At  $.621  each,  how  many  pairs  of  gloves  can  be  bought 
for  1120? 

10.  A  man  bought  potatoes  at  f  .62^  a  bushel  and  sold  them 
at  $  .87-|  a  bushel.  His  profit  was  1 160.  How  many  bushels 
were  sold  ? 

11.  A  dealer  spent  f  120  for  chickens,  and  the  same  amount 
for  ducks.  The  chickens  cost  him  16|  cents,  and  the  ducks 
12|^  cents  a  pound.  How  many  more  pounds  of  ducks  than 
chickens  did  he  purchase  ? 

SPECIAL  CASES   IN  MULTIPLICATION 

149.     I.    To  multiply  a  number  by  10  or  a  power  of  10. 

Each  removal  of  a  figure  one  place  to  the  left  multiplies  its 
value  by  10. 

Therefore,  if  the  multiplicand  is  an  integer,  annex  as  many 
ciphers  as  there  are  ciphers  in  the  multiplier ;  if  the  multipli- 
cand is  a  decimal,  move  the  decimal  point  as  many  places  to 
the  right  as  there  are  ciphers  in  the  multiplier. 

This  is  the  same  as  moving  all  the  figures  to  the  left. 

II.    To  multiply  a  number  by  25. 

25  =  100  -^  4 


58  GRAMMAR  SCHOOL   ARITHMETIC 

« 
Therefore,  multiply  the  given  number  by  100  and  divide  the 
product  by  4.     (Apply  Case  I  in  multiplying  by  100.) 

III.  To  multiply  a  number  by  125. 

125  =  1000  -f-  8 

Therefore,  multiply  the  given  number  by  1000  and  divide 
the  product  by  8.     (Apply  Case  I  in  multiplying  by  1000.) 

IV.  To  multiply  a  number  by : 

a.  .33|^,  multiply  the  given  number  by  \, 

h.  .25,  multiply  the  given  number  by  \, 

c,  .16|,  multiply  the  given  number  by  i. 

d,  .14f,  multiply  the  given  number  by  \, 

e,  .125,  multiply  the  given  number  by  J. 

V.  To  multiply  a  number  by  99. 

99  =  100-1 

Therefore,  multiply  the  given  number  by  100  and  subtract 
the  multiplicand  from  the  product  thus  obtained. 
How  can  we  multiply  a  number  by  999  ? 

VI.  To  multiply  by  a  number  having  one  or  more  ciphers  at 
the  right. 

Multiply  by  the  significant  figures  of  the  multiplier,  and 
annex  to  the  product  thus  obtained,  as  many  ciphers  as  there 
are  in  the  multiplier.     Explain. 

SPECIAL   CASES   IN  DIVISION 

150.    I.    To  divide  a  number  by  10  or  a  power  of  10. 
Each  removal  of  a  figure  one  place  to  the  right  divides  its 
value  by  10. 


SPECIAL  CASES  IN  DIVISION  59 

Therefore,  if  the  dividend  is  an  integer,  point  off  as  many 
decimal  places  as  there  are  ciphers  in  the  divisor ;  if  the  divi- 
dend is  a  decimal,  move  the  decimal  point  as  many  places  to 
the  left  as  there  are  ciphers  in  the  divisor. 

This  is  the  same  as  moving  all  the  figures  to  the  right. 

II.  To  divide  a  number  by  25. 

25  =  JLo^ 

A  number  divided  by  1|^  equals  the  number  multiplied  by 
Y^-Q-,  or  the  number  multiplied  by  4  and  divided  by  100. 

Therefore,  multiply  the  given  number  by  4,  and  divide  the 
product  by  100.     (Apply  Case  I  in  dividing  by  100.) 

III.  To  divide  a  number  by  125. 

Multiply  the  given  number  by  8,  and  divide  the  product  by 
1000.     Explain. 

IV.  To  divide  a  number : 

a.   By  331,  point  off  two  decimal  places  and  multiply  by  3. 
h.    By  16|,  point  off  two  decimal  places  and  multiply  by  6. 

c.  By  14|,  point  off  two  decimal  places  and  multiply  by  7. 

d.  By  .331  divide  by  f  By  .16|?  By  .125?  By  .14f  ? 
By  2.5?     By  .llj? 

V.  To  divide  by  a  number  with  one  or  more  ciphers  at  the 
right. 

Point  off  in  the  dividend  as  many  decimal  places  as  there  are 
ciphers  at  the  right  of  the  divisor,  then  divide  by  the  remain- 
ing figures. 

Explain. 

In  the  following  examples^  find  results  hy  the  methods  given  in 
sections  149  and  150. 


60  GRAMMAR  SCHOOL   ARITHMETIC 

151.  Oral 

1.  34  X  10  16.  32  X  125  31.  27  x  .331 

2.  305^100  17.  32  X. 125  32.  .16|  x  78 

3.  13x200  18.  14,000^700  33.  48x125 

4.  24  X  25  19.  8100  -h  900  34.  31  -^  25 

5.  .00374  X  1000  20.  28  x  .14f  35.  .24  x  16| 

6.  36x25  21.  42  X. 14  2  36.  2.8-^70 

7.  406  X  100  22.  72  X  .16§  37.  .56  x  .125 

8.  830  ■-  10  23.  11  -  .16f  38.  .025  X  3000 

9.  18x10,000  24.  7-^.125  39.  21x.l4f 

10.  1750-10,000  25.  35  X. 142     '  40.  1.6x25 

11.  48,000-1200  26.  72  X. 125  41.  .8-v-25 

12.  360  X  .331  27.  13-  .16|  42.  80  X  .125 

13.  48  X  .125  28.  180  X  .16f  43.  .008  x  1100 

14.  875-10,000  29.  560  X. 25  44.  .5x500 

15.  700-25  30.  19x3000  45.  .7^.125 

152.  Written 

1.  3.85x15,000  10.  6350-25  18.  88.9x17,000 

2.  572x99  11.  47.832-125  19.  62,408 x. 125 

3.  9.07-^25  12.  83,496-4-4000  20.  80,172 x. 331 

4.  63.47  ^.16|  13.  8397-900  21.  5.07x125 

5.  83.750-^.125  14.  87,416  x.l4f  22.  635^25 

6.  1263^142  15.  5.364  X. 125  23.  4.302 -.16| 

7.  864  X. 125  16.  2397x99  24.  23.8-^125 

8.  9654-. 125  17.  453x999  25.  23.8-f-.125 

9.  4.17  X  .33J 


ACCOUNTS   AND   BILLS  61 

ACCOUNTS  AND  BILLS 

153.  Individuals  or  groups  of  individuals  transacting  business 
with  one  another  are  called  parties  to  the  transactions. 

154.  A  record  of  the  business  transactions  between  two  parties 
is  an  account. 

Merchants  and  others  transacting  any  considerable  amount 
of  business  have  sets  of  books  in  which  accounts  are  kept. 

There  are  various  methods  of  recording  transactions  as  they 
occur,  and  arranging  them  in  the  different  books  to  suit  the 
needs  of  the  business ;  but  it  is  the  general  custom  to  copy  all 
accounts,  finally,  in  a  ledger,  which  shows  in  clear,  concise  form 
the  complete  account  of  each  person,  firm,  or  company  with 
whom  business  is  transacted. 

In  the  ledger,  each  person's  account  is  headed  by  his  name. 
Money  paid,  services  rendered,  and  goods  sold  to  him  are 
entered  in  the  left-hand  or  debit  side  of  the  account. 

Money,  services,  and  goods  received  from  him  are  entered 
in  the  right-hand  or  credit  side  of  the  account. 

Accounts  are  balanced  at  regular  intervals  by  footing  the 
debit  side  and  the  credit  side,  and  subtracting  the  smaller  amount 
from  the  greater.  The  difference,  called  the  balance,  is  then 
entered  on  the  side  having  the  smaller  amount.  This  makes 
the  two  sides  equal,  or  balance,  each  other. 

Horizontal  lines  are  then  drawn  below  the  footings,  and  the 
balance  is  brought  forward  to  begin  the  account  for  a  new 
period. 

The  following  form  represents  the  ledger  account  of  Adolph 
Schiller,  for  October  and  November,  at  a  hardware  store.  The 
number  in  the  column  at  the  left  of  dollars  refers  to  the  page 
of  the  day  book  (the  book  in  which  each  day's  transactions 
are  recorded  as  they  occur)  in  which  the  item  was  first  entered. 


62 


GRAMMAR  SCHOOL   ARITHMETIC 


Dr 

CLcial^ik   ^^itUi. 

Ce 

1907 

1907 

Oct. 

7 

Nails 

6 

$5 

75 

Oct. 

20 

Locks 

49 

11 

75 

11 

Doors 

32 

18 

50 

28 

Cash 

54 

50 

19 

Door  trimmings 

48 

7 

48 

31 

Balance 

41 

48 

25 

Windows 

51 

61 
93 

50 
23 

93 

23* 

1907 

Nov. 

1 

Bal.  brought  for'd 

41 

48 

Nov. 

8 

Cash 

58 

20 

10 

White  lead 

60 

7 

40 

15 

Labor 

63 

2 

50 

17 

Shovel 

65 

75 

Note.  —  Many  bookkeepers  omit  from  the  ledger  the  words  describing 
the  articles  bought  and  sold,  as  nails,  locks,  etc.,  leaving  those  columns 
blank.     This  practice  is  increasing. 

Copy  Mr.  Schiller's  account  for  November ;  balance  it,  and 
make  the  proper  entry  to  begin  the  account  for  December. 

At  the  time  of  balancing  an  account,  it  is  customary  to  send 
to  the  debtor  a  copy  of  the  account  for  the  period  for  which  the 
balance  is  made.  This  is  called  a  bill  or  statement.  Many 
business  houses  send  monthly  statements  to  their  customers. 
Some  business  houses  send  a  bill,  or  invoice,  as  it  is  called,  with 
each  list  of  goods  sold. 

155.  The  party  who  sells  the  goods  is  the  creditor;  the  party 
tvho  purchases  the  goods  is  the  debtor. 

In  common  usage,  the  term  debtor  means  any  one  who  owes  a 
debt^  and  the  term  creditor  means  any  one  to  whom  a  debt  is  owed. 

156.  A  bill  may  be  defined  as  follows : 

A  formal  statement  of  a  debtor  s  account,  or  of  goods  sold,  services 
rendered,  or  cash  paid,  made  out  by  the  creditor  and  presented  to 
the  debtor,  is  a  bill. 

A  bill  should  always  contain  these  things : 
1.    The  time  and  place  of  making  out  the  bill. 


ACCOUNTS   AND   BILLS 


63 


2.  The  debtor's  name  and  address. 

3.  The  creditor's  name  and  address. 

4.  A  list  of  the  items — that  is,  the  goods  sold,  money  paid, 
or  services  rendered,  with  the  amount  of  each  item. 

5.  The  date  of  each  transaction,  if  any  of  them  occur  at  any 
other  time  than  that  of  making  out  the  bill. 

6.  The  amount,  or  footing,  of  the  bill. 

157.  When  a  bill  is  paid,  the  creditor  receipts  the  bill  by 
writing  at  the  bottom,  "Received  Payment,"  followed  by  the 
date,  and  his  own  name.  This  shows  that  the  bill  has  been 
paid.     The  debtor  keeps  the  receipted  bill.     Why  ? 

Sometimes  a  clerk,  an  agent,  or  a  bookkeeper  of  the  creditor 
receives  the  money  for  payment  of  a  bill.  He  should  then 
write  the  creditor's  name  under  the  words  "Received  Pay- 
ment," and  under  the  creditor's  name,  his  own  name  or  initials. 

158.  The  following  forms  illustrate  some  of  the  ways  in 
which  bills  are  made  out: 


FORM  I 

Mrs.  John  Doe 

1421  West  Street, 

Boston,  Mass. 


Boston,  October  I,  1908. 


BauBhtot  IV.  §♦  0kartis  $?  Compatig 


140  TREMONT  STREET  BOSTON 


98 
128 


SEPT. 


1  GLOVES 

1  1/4  VEILING 

1  1/4  " 


.25 
50 


4.00 

4.00 

.31 

.63 

.94 

4.94 

64 


GRAMMAR   SCHOOL   ARITHMETIC 


SHEET  IRON  PIPING  A  SPECIALTY 
ORDER  NO.     51673 

Mr.  John  R.  McKavney 


FORM    2 

SHEET  METAL  WORK  OF  EVERY  DESCRIPTION 

PITTSBURG,  PA.       Aug.  16,   1908 
2528  Penn  Ave. 


BouGHTOF  Grant  C.  Nobbs, 

SHEET  METALWORKAND  HARDWARE 


PENN  PERFECT  FURNACES. 

SALESROOM  AND   WAREHOUSE 

2623  AND  2625  PENN  AVE. 


TIN  ROOFING. 


STOVES  AND  HOUSEFURNISHINO  GOODS. 


BOTH  PHONES 


OFFICE  AND   WORKS 
2520  AND  2522  SMALLMAN  ST. 


TERMS 

30  days 

i 

Doz.  8"  Hinge  Hasps 

40  -  10? 

@  1.05 

53 

29 

10 

Gr.  i-7  Screws 

87i  -  5% 

@   .90 

9 

00 

1 

08 

i 

2 

Doz.  #338  Half  Hatchets 

@  6.00 

3 

00 

3 

"  #1  Sledge  Handles 

@  1.10 

3 

30 

2 

Kegs  Common  Nails 

@  2.10 

4 

20 

1 

Doz.  Kules  #68 

@   .95 

95 

2 

Stanley  Planes 

@   .94 

1 

88 

3- 

4 

Doz.  Niagara  Handled  Axes 

@  6.75 

5 

06 

19 

76 

159.  Oral 

1.  Name  the  debtor  and  the  creditor  in  Form  1.    In  Form  2. 
In  P'orm  3.     In  Form  4. 

2.  Which  of  the  forms  contain  both  debit  and  credit  items? 

3.  Which  of  the  forms  contain  items  for  which  bills  have 
been  previously  sent? 

4.  In  Form  5,  what  is  the  amount  of  the  credit  items?    Of 
the  debit  items?    What  is  the  balance? 

160.  Written 

1.    Make  out   the    bill  sent  to  Mr.    Schiller  (see  page   62) 
on  Dec.  1,  1907,  supplying  names,  dates,  and  addresses.     The 


ACCOUNTS   AND  BILLS 


65 


FORM  3 


A.  J.  REACH  CO. 

MAKERS  OF  FINE  SPORTING  GOODS 


Terms  Strictly 
Net  30  Days  or  2^  10  Days 


PHILADELPHIA    8/30/07 


YOUR  ORDER   8/27/07 

Sold  to 

ouRORDEk    8/30/07 


Simmons  Hardware  Co , 


SHIPPED  VIA  St .  Loud  s ,  Mo . 
1   Case  Weight  150  lbs.    Star  Union  Line 


SHIPPED 

20 
1 

6/12 
9/12 


Dz  0      Balls  15.00 

Dz   5A    Catcher's  Mitts  84.00 

Dz   3      First  Baseman's  Mitts   48.00 
Dz   OC   Fielder's  Gloves  30.00 


RECEIVED  PAYMENT 
.....f.  mo.,_/j<p    day.      190,  % 

A.J.  REACH  CO. 


300 

00 

84 

00 

24 

00 

22 

50 

430 

50 


first  debit  item  should  be,  "Account  rendered,  141.48,"  be- 
cause that  was  the  balance  shown  on  the  bill  which  he  re- 
ceived Nov.  1. 

Receipt  the  bill  as  though  you  were  cashier  for  the  creditor. 

Make  out  and  foot  bills  of  the  following  items,  supplying 
dates  and  addresses ;  receipt  them,  either  as  creditor,  or  as  the 
creditor's  agent : 

2.   Bought  by  W.  J.  McDermott  from  Bentley  and  Settle, 

20  bbl.  patent  flour,  85  per  bbl;  2000  lb.  granulated  sugar, 
$5.15  per  hundredweight;  300  lb.  Java  coffee,  22^  per  pound; 
250  lb.  maple  sugar,  14^  per  pound. 

McDermott  has  paid  $125  in  money. 


66 


GRAMMAR  SCHOOL   ARITHMETIC 


STATEMENT 


FORM   4 
Philadelphia,  Pa.,      /V^^^^/;  190%- 

A.J.  Reach  Company 

TULIP  AND  PALMER  STREETS 


^^rM^A^. 


M^ 


s^^SZ^i 


TERMS:- 

-NET  CASH  30  DAYS 

Amount  Rendered 

:ai- 

/ 

To  lliase.,.a8  per  1>ill  rendered 

.?/ 

f/ 

/"f 

//,f 

'W 

^.^ 

/.0.7 

op 

J?,5' 

7.9'? 

3J} 

„                              / 

-^/ 

00^ 

rTv 

/^. 

/^..      "7??^^.. 

p 

{0,9 

AT 

''^     Cl.^J^ 

-Of 

D.8- 

. 

D{n 

}l                   » 

/(7t) 

OV 

/ 

^ 

^ 

/.?/ 

^  /^ 

/ 

j-^ 

/ 

^7fo 

.2^ 

3.    A.  Walrath  sold  to  Donald  Anderson, 

5  lb.  rice  at  9  /. 

4  dozen  eggs  at  21  ^. 

2  brooms  at  35  ^. 

18  lb.  chicken  at  22  ^, 

2  bu.  new  potatoes  at  35/  per  peck. 

8  lb.  tomatoes  at  13/  per  pound. 


ACCOUNTS   AND   BILLS 
FORM   5 


67 


SOUTH  SAUNA  «.  JEFFERSON  STS. 


c^ cM'  I   /f  Y 


■^^-m 


fos(ik4'e^  /:i^- 


THIS  BILL    WILL  BC  CHECKED   BY  US  AS  PEfirECTLY  COPRECT  UNLESS  KEPOKTCD    OTHERWISE  WmHIN   TEN  DAYS 


Q/I/n 

(2,^,  ?j^. 

C 

S^ 

^U^u,   V 

1 

^^/7yf^                                                      /V 

^^ 

1 

■'^..^^A^JrA 

-/ 

7^ 

l/:<5/ 

n 

.^^:S^r/^                                     cr 

/ 

n 

.^^  «a                     ■^: 

^ 

v^r 

;-     ,3 

C>\f 

/p^\ 

// 

yf 

/foy 

/^    -m^^^VA 

//l^ 

>i^ 

/  /a/^<     ly^^ 

/ 

<y<j 

<^s^ 

M 

ti_ 

4.   D.  M.  Edwards  sold  to  Henry  Fenner, 
June    1,  Account  rendered,  f  15. 
June  26,  1  pattern  $.15. 

21  yd.  dimity,  12|  ^  per  yd. 
July    1,  4  yd.  chiffon,  50^  per  yd. 

2  doz.  buttons,  15^  per  doz. 

41yd.  braid  at  $.33. 

Credit 
July    6,  2  yd.  chiffon  at  50^. 
1  pr.  gloves,  il.25. 
Cash,  $7.75. 


68  GRAMMAR   SCHOOL  ARITHMETIC 

5.    Debtor,  Miss  Margaret  Maddox ; 
Creditor,  H.  G.  Stone  &  Son. 

Account  rendered,  112.35. 
24  yd.  lace  at  25)^. 

2  spools  twist  at  |.05. 

3  doz.  yd.  lace  at  il.25. 
6f  yd.  net  at  $.621. 

6f  yd.  linen  at  |. 621 

Credit 
Cash,  $10. 

REVIEW  AND  PRACTICE 
161.    Oral 

1.  Read  300.00300;    2000.002;    860.0860;    CXIV ;   XLIV ; 
MCMIX. 

2.  Find  the  change  from  $1  for  $.28;   $.36;   $.71;   $.81; 
$.53;  $.m;  $.17. 

3.  75  is  how  many  times  25?     If  25  crates  of  oranges  cost 
$90,  what  Avill  75  crates  cost  at  the  same  price  per  crate  ? 

4.  Add   in   the    easiest   way    28   and   45;    63  and  89;    16 

and  87. 

5.  Name  six  aliquot  parts  of  $1. 

6.  Using  aliquot  parts,  find 

a.  The  cost  of  32  packages  of  hominy  at  12|^^  a  package. 

h.   The  quantity  of  dates  that  $10  will  buy  at  6J^  per  pound. 

c.    The  number  of  sheets  of  sandpaper  that  can  be  bought 
for  $  2,  at  8^  ^  per  dozen  sheets. 

7.  Annexing   four   ciphers   to   an   integer  affects  its  value 
how  ? 


REVIEW   AND   PRACTICE  69 

8.  Name  two  composite   numbers   that  are  prime  to  each 
other. 

9.  What  is  the  smallest  number  that  will  exactly  contain 
18  and  27? 

10.  What  is  the  greatest  number  that  will  exactly  divide  60, 
15,  and  90? 

11.  Name  the  prime  numbers  between  80  and  115. 

12.  The  product  of   two  numbers  is  20,000.     One  of   the 
numbers  is  50.     What  is  the  other  number  ? 

13.  I  is  the  product  of  5  and  what  other  number  ? 

14.  1200  is  the  product  of  30,  4,  and  what  other  number  ? 

15.  1.824  =  1824-? 

16.  .0375  =  375^? 

17.  93  -  3  X  11  +  200  -V-  5  =  ? 

18.  (93  -  3)  X  (38 -28) -^(5x18)  =  ? 

19.  Reduce  to  simplest  form  :    || ;  ^f ;  f  f- ;  If  ;  -fl ;  yf  • 

20.  Reduce  |,  |,  -^^  and  |  to  fractions  having  a  common 
denominator. 

21.  From  181  take  7f . 

22.  Tell  which  of  the  following  fractions  cannot  be  reduced 
to  exact  decimals,  and  why ;    y^y,  |-,  ||,  3^5,  2^0^,  |^. 

23.  Multiply  31  by  99. 

24.  Divide  7000  by  25. 

25.  How  can  you  tell,  without  actual  trial,  that  742  will  not 
exactly  divide  1,834,659  ? 


70  GRAMMAR   SCHOOL   ARITHMETIC 

26.  How  may  we  know,  without  actual  trial, 

a.  That  8  will  not  exactly  divide  4,379,624,700  ? 
h.  That  5  will  not  exactly  divide  3,079,628  ? 

c.  That  9  will  exactly  divide  2,405,376  ? 

d.  That  25  will  exactly  divide  397,400  ? 

27.  There  are  seven  decimal  places  in  a  product  and  three 
decimal  places  in  one  of  its  two  factors.  How  many  decimal 
places  are  there  in  the  other  factor  ? 

28.  The  numerator  of  a  fraction  is  which  term  in  division  ? 
The  denominator  ?     The  value  of  a  fraction  ? 

29.  A  certain  number  containing  five  decimal  places  is  the 
product  of  three  factors.  One  of  its  factors  contains  two  deci- 
mal places,  and  another  factor  three  decimal  places.  The  re- 
maining factor  contains  how  many  decimal  places  ? 

30.  Make  a  problem  which  can  be  solved  by  the  use  of 
aliquot  parts. 

31.  What  number  is  the  product  of  all  the  common  prime 
factors  of  84  and  132  ? 

32.  One  of  the  school  buildings  in  a  certain  city  was  heated 
by  150  tons  of  coal,  costing  810  dollars.  At  the  same  price  per 
ton,  what  was  the  cost  of  the  coal  for  a  school  that  required 
75  tons  ? 

33.  3  X  19  -  7  +  150  ^  2  =  ? 

34.  The  cost  of  a  number  of  horses  is  a  product.  The 
number  of  horses  is  one  factor.     What  is  the  other  factor  ? 

35.  480  is  six  times  what  number  ?  Which  of  these  numbers 
is  a  product  ?     The  number  to  be  found  is  what  term  ? 

36.  32  is  .16  of  what  number?  32  is  which  term  in  multi- 
plication ?     Which  terms  are  .16  and  the  number  to  be  found  ? 


REVIEW  AND   PRACTICE  71 

37.  The  yearly  wages  of  36  men  in  a  factory  amount  to 
$28,800.     At  the  average  wages,  what  do  12  men  receive  ? 

38.  A  farmer  shipped  32  cans  of  milk  to  the  city  in  one 
week,  each  can  containing  40  quarts.  How  many  gallons  did 
he  ship  ? 

39.  .331  of  $18  =  ?  $6  =  .331  of  what  ?  $Q  are  how  many 
hundredths  of  1 18? 

40.  A  seamstress  buys  a  sewing  machine  for  $55.  If  she 
pays  $25  at  the  time  of  purchase,  and  $5  every  month  there- 
after, in  how  many  months  will  she  finish  paying  for  the 
machine  ? 

41.  How  may  we  tell  at  a  glance  that  6  will  not  exactly 
divide  176,435  ?     That  6  will  exactly  divide  933,012  ? 

42.  I  of  the  length  of  a  trench  is  60  feet.  What  is  |  of  its 
length  ? 

43.  24  will  exactly  divide  a  certain  number.  Name  six  other 
numbers  that  will  exactly  divide  that  number. 

44.  4.5  yards  of  lace  cost  $2.70.  What  is  the  cost  of  1.5 
yards  of  the  same  lace  ?     Of  1  yard  ? 

45.  1946-^19.46  =  ? 

46.  Read  2.00500;  300,083.383;  .62550;  62,500.00050. 

47.  Read  CDLXXV;  CCCXCIII;  MCXLIV;  GUI; 
CXIV;  XCVII. 

48.  a.  I  of  f  =  ? 

h.  1^  is  I  of  what  number  ? 
c.  What  part  of  f  is  ^  ? 

49.  At  a  fruit  stand,  peaches  are  marked  "4  for  5  cents." 
What  does  the  dealer  receive  for  36  peaches  ? 

50.  Divide:  a.  2496  by  10,000  ;  h.  36.16  by  .04 ;  c.  13  by  125 ; 
d.  5600  by  400. 


72  GRAMMAR  SCHOOL  ARITHMETIC 

51.  Determine  which  of  the  following  numbers  are  prime 
and  which  are  composite : 

91;  97;  111;  203;  37,564,296;  131;  141;  113;  109. 

52.  How  may  we  test  the  accuracy  of  our  work  in  addition  ? 
In  subtraction  ?     In  multiplication  ?     In  division  ? 

162.    Written 

In  examples  1-5  find  the  sums  and  test  hy  adding  in  a  different 
order.      Time  yourself. 


1. 

2. 

3. 

4. 

5. 

385.21 

15.182 

92.75 

837. 

99.37 

46.83 

619.83 

689.98 

.96 

48.69 

795.467 

50.70 

7.42 

43.82 

372.918 

18.23 

912.183 

9.87 

4.79 

72.75 

963.542 

28.764 

48.136 

10.68 

4.681 

795.087 

783.908 

7.091 

5.30 

.37 

32.145 

58.392 

36.98- 

12.98 

.984 

819.768 

75.64 

74.132 

4.672 

98.307 

73.242 

9.728 

8.007 

.89 

8.137 

53.718 

12.34 

2.19 

3.765 

4.90 

910.763 

90.806 

63.981 

48.92 

25.36 

42.86 

9.173 

3.42 

7.96 

7.008 

8.51 

20.304 

7.895 

12.834 

.93 

793.916 

58.79 

9.86 

.098 

24.135 

213.804 

9.309 

57.713 

1.39 

.86 

67.51 

864.23 

8.88 

4,06 

7.19 

In  examples  6-15  subtract  and  test  your  worh^  timing  yourself: 

6.    38700.5  7.    17934.68  8.    $7000.53 

498.499  279.69  909.44 


REVIEW  AND  PRACTICE  73 


9.  1801010.02 

11.  151000.001 

13.  240.50 

1900.92 

1900.92 

39.49 

10.  .13400.75 

12.  28037.6 

14.  23037.644 

2896.075 

280.799 

280.799 

15.  Find  the  difference  between  24007.901  and  980.89. 

In  examples  16-21  find  results^  and  test  your  work  hy  the  re- 
verse operation: 

16.  3.07x51.8        18.    92.007x380      20.    2133.854-^5.08 

17.  7968^5.38        19.    8.05x39.8         21.    83412-6000 

22.  Multiply  837  by  12,  and  test  your  work  by  addition. 

In  examples  23-40  perform  the  indicated  operations  in  the 
shortest  way: 

23.  287x125  29.  90,876  X. 331  35.  48.35x7000 

24.  876-^25  30.  8642 -- 16f  36.  859x1.25 

25.  563x99  31.  50.74-^125  37.  2100 -^  70,000 

26.  481x2500  32.  39.72x99  38.  548  x  33 J 

27.  3074 -f- 125  33.  47.012  X. 25  39.  7867-^16f 

28.  4.207x25  34.  88.7 ^.14f  40.  6570  x  .25 

41.  A  miller  ground  .25  of  a  load  of  corn  into  meal,  and 
cracked  .35  of  the  load  for  chicken  feed.  There  remained  360 
bushels.     The  carload  consisted  of  how  many  bushels  ? 

42.  A  man  who  owned  .375  of  a  ship  sold  ^  of  his  share  for 
$24,000.     What  was  the  entire  value  of  the  ship? 

43.  Express  in  words,  and  in  Roman  numerals,  the  number 
of  the  present  year. 


74  GRAMMAR   SCHOOL   ARITHMETIC 

44.  A  music  dealer  marked  a  piano  at  1750  and  sold  it  for 
.83|-  of  the  marked  price.     How  much  did  he  receive  for  it? 

45.  A  man  owns  three  houses.  He  rents  the  first  for  $276 
a  year,  the  second  for  $450,  and  the  third  for  |  as  much  as  he 
receives  for  the  first  two.  How  much  rent  does  he  receive  in 
5  years  ? 

46.  A  monthly  magazine  averages   92   pages  of   advertise 
ments  each  month.     It  receives  $276,000  a  year  for  advertise- 
ments.   What  is  the  average  cost  of  one  page  of  advertisements 
for  one  month  in  this  magazine  ? 

47.  The  steamship  Lusitania^  during  one  trip,  consumed  50 
tons  of  coal  per  hour.  At  this  rate,  how  many  tons  will  she 
consume  on  a  voyage  lasting  four  days  and  twenty  hours  ? 

48.  a.  Mr.  Rogers  uses  ^  of  his  yearly  income  for  household 
expenses  and  |  of  the  remainder  for  his  son's  tuition.  What 
fraction  of  his  income  is  left  ? 

h.    If  1 660  are  left,  how  much  does  the  son's  tuition  cost  ? 

49.  What  must  be  added  to  882^  to  obtain  121^^6  ? 

50.  A  custom  miller  used  to  take  \  of  the  grain  as  toll  to 
pay  him  for  grinding  the  remainder  of  it.  He  took  376J  lb. 
for  grinding  a  load  of  wheat.  If  a  bushel  of  wheat  weighs 
60  lb.,  how  many  bushels  did  the  load  of  wheat  contain  ? 

51.  Make  and  solve  : 

a.  A  problem  that  requires  addition  and  subtraction. 

b.  A  problem  that  requires  addition  and  multiplication. 

c.  A  problem  that  requires  multiplication  and  division. 

52.  Make  out,  foot,  and  receipt  a  bill  containing  three  debit 
items  and  one  credit  item. 

53.  Make  and  solve  a  problem  that  requires  you  to  find  the 
least  common  multiple. 


ARTICLES   SOLD   BY   THE   THOUSAND  75 

ARTICLES  SOLD    BY  THE   THOUSAND,  HUNDRED,   OR 
HUNDREDWEIGHT 
163.     Written 

1.  What  is  the  cost  of  8975  bricks  at  i  7  per  M.?     (M.  stands 
for  1000.) 

8975  =  8.975  M. 

Since  1  M.  costs  1 7,  8.975  M.  cost  8.975  x  |7,  or  $ Ans. 

2.  What  must  be  paid  for  980  soapstone  pencils  at  f  .30  per 
C.  ?     (C.  stands  for  100.) 

980  =  9.80  C. 

Since  1  C.  costs  |.30,  9.80  C.  will  cost  9.80  x  |.30,  or  | Ans. 

3.  Find  the  cost  of  1550  lb.  of  new  buckwheat  flour  at  12.50 
per  hundredweight  (100  pounds). 

1550  lb.  =  15.50  hundredweight. 

Since  1  cwt.  costs  ^2.50,  15.50  cwt.  cost  15.50  x  $2.50,  or  $ Ans. 

Note.  —  In  final  results,  a  fraction  of  a  cent,  equal  to  or  greater  than  i  cent, 
is  counted  a  whole  cent.     A  fraction  which  is  less  than  |  cent  is  dropped. 

4.  Find  the  cost  of  each  of  the  following  items : 
a.    83,900  bricks  at  17.80  per  M. 

h.    8950  lb.  sugar  at  $4.95  per  C. 
c.    1550  asparagus  roots  at  $.95  per  C. 
6?.   10,000  laths  at  $.45  per  C. 
e.    12,500  paper  butter  trays  at  $.40  per  M. 
/.   25  barrels  of  granulated  sal  soda,  each  barrel  containing 
325  lb.,  at  $.90  per  hundredweight. 
g.    25,600  cakes  of  naphtha  soap  at  $3.25  per  C. 
h.    1700  cubic  feet  of  gas  at  $.95  per  M. 


T6  GRAMMAR  SCHOOL  ARITHMETIC 

DENOMINATE  NUMBERS 

164.  A  number  that  is  composed  of  units  of  weight  or  measure 
is  a  denominate  number;  e.g.  5  lb.,  7  rd.,  6  hr.  3  min.  45  sec. 

165.  The  name  of  a  unit  of  weight  or  measure  is  a  denomina- 
tion; e.g.  ounce,  square  foot,  minute. 

166.  A  denominate  number  that  is  expressed  in  two  or  more 
denominations  is  a  compound  number ;  e.g.  1  yd.  2  ft.  7  in. ; 
2  lb.  14  oz. 

167.  A  number  that  is  expressed  in  but  one  hind  of  units  is  a 
simple  number  ;  e.g.  3  days,  8  cents,  19  pounds,  125. 

168.  TABLE   OF  LIQUID  MEASURE 

4  gills  (gi.)  =  1  pint  (pt.). 
2  pints  =  1  quart  (qt.). 

4  quarts        =  1  gallon  (gal.). 

Oil,  vinegar,  molasses,  and  other  liquids  are  shipped  in  barrels 
or  casks  of  various  sizes.  But  for  the  purpose  of  indicating  the 
capacities  of  vats,  tanks,  reservoirs,  etc.,  31 1^  gallons  are  called 
a  barrel  (bbl.)  and  63  gallons  a  hogshead  (hhd.). 

169.  TABLE   OF  DRY  MEASURE 

2  pints  (pt.)  =  1  quart  (qt.). 
8  quarts  =  1  peck  (pk.). 

4  pecks  =  1  bushel  (bu.). 

170.  TABLE   OF  AVOIRDUPOIS  WEIGHT 

16  ounces  (oz.)  =  1  pound  (lb.). 
2000  pounds     =  1  ton  (T.). 
2240  pounds     =  1  long  ton. 
100  pounds     =  1  hundredweight  (cwt.). 


DENOMINATE   NUMBERS  77 

The  term  hundredweight  is  used  less  than  formerly,  although 
its  value  (100  lb.)  is  still  taken  as  a  unit  in  quoting  freight 
rates  and  prices  of  various  articles,  when  the  quantity  used 
makes  this  a  convenient  unit  of  weight. 

The  long  ton  is  used  in  wholesaling  certain  mining  products. 

The  ton  of  2000  lb.  is  sometimes  called  a  short  ton. 

171.  TABLE   OF  TROY  WEIGHT 

24  grains  (gr.)     =1  pennyweight  (pwt.). 

20  penny  weights  =  1  ounce  (oz.). 

12  ounces  =1  pound  (lb.). 

These  weights  are  used  in  weighing  gold,  silver,  and  some 
jewels.  To  get  an  idea  of  the  weight  of  a  grain,  think  of  the 
weight  of  a  grain  of  wheat  or  rice. 

172.  TABLE  OF  APOTHECARIES*  WEIGHT 

20  grains  (gr.)  =  1  scruple  (sc.  or  3). 
3  scruples         =  1  dram  (dr.  or  3). 
8  drams  =  1  ounce  (oz.  or  %), 

This  table  is  used  by  druggists  and  physicians  in  compound- 
ing medicines;  but  medicines  are  bought  and  sold  by  Avoir- 
dupois weight,  except  in  quantities  smaller  than  one  ounce. 

173.  Druggists  use  a  iQvm  fluid  ounce,  which  is  not  a  meas- 
ure of  weight,  but  of  capacity,  and  is  equal  to  -f^  of  a  pint. 
Thus,  a  2-ounce  bottle  is  a  bottle  that  holds  |^  of  a  pint  of  any 
liquid  regardless  of  its  weight. 

174.  TABLE   OF  LINEAR  MEASURE 

12  inches  (in.)  =  1  foot  (ft.). 
3  feet  =  1  yard  (yd.). 

8/?»r1    ->"'^<"';> 

320  rods  =1  mile  (mi.). 


78  GRAMMAR  SCHOOL  ARITHMETIC 

175.  TABLE  OF   SURVEYORS'  LONG  MEASURE 

7.92  inches  =  1  link  (li.). 
100  links    =1  chain  (ch.). 
80  chains  =  1  mile  (mi.). 

This  table,  formerly  used  by  surveyors  in  measuring  land, 
should  be  learned  and  remembered,  because  descriptions  of 
land  in  the  public  records  of  deeds  and  mortgages  are  largely 
made  in  the  denominations  of  this  measure. 

176.  TABLE  OF  SURFACE  MEASURE 

144  square  inches  (sq.  in.)  =  1  square  foot  (sq.  ft.). 

9  square  feet  =  1  square  yard  (sq.  yd.). 

30 J  square  yards        "  =  1  square  rod  (sq.  rd.). 

160  square  rods  =  1  acre  (A.). 

640  acres  =  1  square  mile  (sq.  mi.). 

177.  TABLE  OF  SURVEYORS'  SQUARE  MEASURE 

625  square  links    =  1  square  rod. 
16  square  rods     =  1  square  chain. 
10  square  chains  =  1  acre. 

This  table,  like  that  of  Surveyor's  Linear  Measure,  is  used 
in  public  records,  chiefly. 

178.  TABLE   OF  VOLUME  MEASURE 

1728  cubic  inches  (cu.  in.)  =1  cubic  foot  (cu.  ft.). 
27  cubic  feet  =  1  cubic  yard  (cu.  yd.). 

179.  TABLE  OF  COUNTING 

12  =1  dozen  (doz.). 

12  doz.  =  1  gross. 
20  =1  score. 


DENOMINATE   NUMBERS  79 

180.  TABLE  OF  TIME 

60  seconds  (sec.)  =  1  minute  (min.). 

60  minutes  =  1  hour  (hr.). 

24  hours  =1  day  (da.). 

7  days  =  1  week  (wk.). 

52^  weeks  =  1  common  year  (yr.). 

52|^  weeks  =  1  leap  year. 

365  days  =  1  common  year. 

366  days  =  1  leap  year. 

Ten  years  are  called  a  decade,  and  one  hundred  years  make 
a  century^  but  these  terms  are  not  used  in  arithmetical  calcula- 
tions. ^ 

The  four  thirty-day  months  may  be  remembered  easily  by 
the  following  old  rhyme  : 

"  Thirty  days  hath  September, 
April,  June,  and  November." 

February  has  28  days,  with  29  in  leap  year.  The  other 
months  have  31  days. 

The  exact  length  of  the  solar  year,  that  is,  the  time  of  one 
revolution  of  the  earth  around  the  sun,  is  365  days  5  hours  48 
minutes  and  46  seconds,  or  nearly  365|  days.  By  adding  one 
day  to  the  365  every  fourth  year,  too  much  time  is  added. 
This  is  corrected  by  counting  every  centennial  year  as  a  com- 
mon year,  except  when  its  number  is  divisible  by  400.  The 
year  1900,  therefore,  was  not  a  leap  year,  although  its  number 
was  divisible  by  4.  . , 

181.  TABLE   OF  PAPER  MEASURE 

24  sheets  =  1  quire. 
20  quires  =  1  ream. 


80  GRAMMAR   SCHOOL   ARITHMETIC 

The  terms  bundle  (2  reams)  and  hale  (5  bundles)  are  seldom 
used.  The  denomination  quire  is  used  mostly  in  measuring  the 
finer  grades  of  writing  paper.  Wrapping  paper  is  sold  by  the 
pound  or  by  the  thousand  sheets.  Many  kinds  of  paper  are  sold 
in  packages  of  five  hundred  or  one  thousand  sheets.  Packages 
of  five  hundred  sheets  are  sometimes  called  reams. 

182..  TABLE  OF  UNITED   STATES  MONEY 

10  mills  =  1  cent. 
10  cents  =  1  dime. 
10  dimes  =  1  dollar. 

The  gold  coins  of  the  United  States  are  the  $5,  flO,  and 
$20  pieces,  once  called  the  half  eagle,  eagle,  and  double  eagle. 
Gold  dollar  coins  are  not  in  general  circulation,  although  a  few 
of  them  have  been  made. 

The  silver  coins  are  the  dollar,  half  dollar,  quarter  dollar, 
and  dime.  Silver  half  dimes  are  no  longer  coined.  Most  five- 
cent  pieces  are  made  of  nickel.  Most  1-cent  pieces  are  made 
of  bronze,  though  some  nickel  and  copper  cents  are  in 
circulation. 

The  mill  is  not  coined. 

183.  TABLE  OF  ENGLISH  MONEY 
4  farthings  (far.)  =  1  penny  (c?.). 

12  pence  =  1  shilling  (s.). 

20  shillings  =  1  pound  (£). 

Farthings  are  not  coined,  and  are  commonly  written  as 
fractions  of  a  penny. 

184.  TABLE  OF  FRENCH  MONEY 

100  centimes  =  1  franc. 


ARC   AND  ANGLE  MEASURE  81 

185.  TABLE   OF   GERMAN  MONEY 

100  pfennigs  =  1  mark. 

186.  The  denominations  of  Canadian  money  are  like  those  of 
the  United  States. 

187.  TABLE  OF  ARC  AND  ANGLE  MEASURE 

60  seconds  (^')  =  1  minute  ('). 
60  minutes  =  1  degree  (°). 
An  arc  of  360°  =  1  circumference. 

188.  The  difference  in  direction  of  two  lines  that  meet  is  an 
angle;   e.g.  ^^^ 

Angles 

189.    The  lines  that  meet  to  form  an  angle  are 
-^   the  sides  of  the  angle. 

Lines  are  read  by  means  of  letters   placed 
^  at    their    extremities.       Angles    are    read   by 

means  of  letters  placed  at  the  extremities  of  their  sides. 
In  the  angle  ABQ  the  lines  AB  and  ^(7  are  the  sides. 

190.  The  sum  of  all  the  angles  that  can  he  formed  around  a 
point  in  a  plane  is  360°. 

191.  A  plane  figure  hounded  hy  a  curved  line., 
every  point  of  which  is  equally  distant  from  a 
point  within  called  the  center^  is  a  circle. 

192.  The  houndary  line  of  a  circle  is  its 
circumference. 

193.  Any  part  of  a  circumference  is  an  arc. 


82 


GRAMxMAR   SCHOOL   ARITHMETIC 


194.    The  number  of  degrees  in  an  arc  is  always  the  same  as  the 
number  of  degrees  in  the  angle  at  the  center^  whose  sides  meet  the 
A  extremities  of  the  arc^  thus  ; 

The  angle  AOB  is  \  the  sum  of  all  the 
angles  at  the  center,  or  90°.  The  arc  AB  is 
\  of  the  circumference,  or  90°.  Can  you  tell 
the  number  of  degrees  in  the  arc  BU  ?  In  the 
Single  UOB? 

Angles  are  measured  by  a  protractor,  an  instrument  made 
of  metal,  with  degrees  marked  and  numbered  as  shown  below. 


.z' 


,^'M 


,/ 


Fig.  1 


Fig.  2 


^^H 


To  measure  the  angle  AOO,  the  protractor  is  placed  as  indi- 
cated, so  that  the  center,  0,  of  the  protractor,  coincides  with 


ARC   AND   ANGLE   MEASURE  83 

the  vertex,  0,  of  the  angle,  and  the  sides  A  0  and  CO  take  the 
positions  indicated.  .  Th6  scale  on  the  protractor  indicates  that 
the  angle  A  00  is  an  angle  of  20°. 

Notice  that  the  length  of  the  sides  does  not  affect  the  size  of 
the  angle.  They  may  be  prolonged  indefinitely  without  chang- 
ing the  angle. 

State  the  number  of  degrees  in  each  of  the  following  angles 
as  indicated  by  the  protractor  in  Fig.  3 : 


a. 

Angle  A  OB 

e. 

Angle  MOK 

i. 

Angle  BOB 

b. 

Angle  A  OB 

/. 

Angle  MOII 

J- 

Angle    COB 

e. 

Angle  AOE 

^• 

Angle  BOK 

k. 

Angle  BOB 

d. 

Angle  HOK 

h. 

Angle   OOE 

I 

Angle   COS 

195.  An  angle  of  90°  is  a  right  angle. 

196.  An  angle  that  is  greater  than  a  right  angle  is 
an  obtuse  angle. 

197.  An  angle  that  is  less  than  a  right  angle  is  an 
acute  angle. 

In  Fig.   3,  what  kind  of  angle   is    angle  AOC? 
Angle  AOB?     Angle   AOB?     Angle   AOB?    Angle    BOB? 

198.  MISCELLANEOUS   DENOMINATIONS 

6  feet  =  1  fathom,  used  in  measuring  the  depth  of  the  water 
at  sea. 

40  rods  =  1  furlong. 

4  inches  =  1  hand,  used  in  measuring  the  height  of  horses. 

1.15  common  or  statute  miles  =  1  nautical  mile,  or  knot, 
used  in  measuring  distances  at  sea  and  the  speed  of  vessels. 
The  nautical  mile  is  assumed  at  6086.07  feet,  or  1.152664 
statute  miles,  by  the  United  States  Coast  Survey.  For  ordi- 
nary purposes  of  computation,  however,  1.15  is  sufficiently  exact. 


84  GRAMMAR  SCHOOL  ARITHMETIC 

3  nautical  miles  =  1  league. 

640  acres,  or  one  square  mile,  =  1  section  of  land. 
3.2  grains  (approximately)  =  1  carat,  used  in  indicating  the 
weight  of  diamonds  and  other  gems.  The  term  carat  is  also 
used  in  indicating  the  fineness  of  gold.  14-carat  gold,  or  gold 
that  is  14  carats  fine,  is  metal  of  which  l-|  is  pure  gold,  and  H 
is  alloy  (that  is,  harder  metal  mixed  with  the  gold  to  make 
it  more  durable).  The  word  is  sometimes  spelled  karat  and 
jewelers  use  the  abbreviation  k  in  rings  and  other  gold  articles. 
What  is  the  meaning  of  2^-k  gold?  Of  1^-k  gold?  Of  10-k 
gold? 

The  term  perch  is  sometimes  used  to  indicate  (a)  one  rod  in 
length,  or  (5)  one  square  rod  of  land,  or  (c)  a  quantity  of  stone 
or  masonry  1  rod  long,  1|  feet  wide,  and  1  foot  thick,  contain- 
ing 24|  cubic  feet. 

One  hundred  square  feet  of  painting  or  roofing  are  called  a 
square. 

199.  TABLE  OF   EQUIVALENTS 

1  gallon  =  231  cubic  inches. 

1  bushel  =  2150.42  cubic  inches. 

1  pound  Avoir.         =  7000  grains. 

1  pound  Troy  =  5760  grains. 

1  pound  Apoth.       =  5760  grains. 

£  1  (Gt.  Britain)    =  14.8665. 

1  franc  (France)     =  1.193. 

1  franc  (Belgium)  =i.l93. 

1  lira  (Italy)  =.f.l93. 

1  mark  (Germany)  =  | .  238. 

1  yen  (Japan)  =1.498. 

1  ruble  (Russia)      =$.515. 


REDUCTION   OF   DENOMINATE   NUMBERS  85 

The  grain  is  the  same  in  the  three  weights,  Avoirdupois, 
Troy,  and  Apothecaries'.  It  is  obtained  by  taking  a  certain 
fraction  (very  nearly  2^3)  of  the  weight  of  a  cubic  inch  of  dis- 
tilled water  at  its  greatest  density  (39.2°  nearly). 

REDUCTION  OF  DENOMINATE  NUMBERS 

200.  Changing  numbers  to  larger  denominations  is  reduction 
ascending. 

201.  Changing  numbers  to  smaller  denominations  is  reduction 
descending. 

202.  Oral 

1.  Find  the  number  of  cubic  inches  in  one  quart,  liquid 
measure. 

2.  A  fountain  contains  four  barrels  of  water.  How  many 
gallons  does  it  contain  ?     How  many  hogsheads  ? 

3.  A  grocer  bought  green  peas  at  two  dollars  a  bushel,  and 
retailed  them  at  ten  cents  a  quart.  What  did  he  gain  on  three 
bushels  ? 

4.  What  is  a  grocer's  profit  on  half  a  ton  of  coffee  bought 
at  f  15  per  hundredweight  and  sold  at  20  cents  a  pound? 

5.  A  gold  dollar  weighs  25.8  grains.  How  many  gold 
dollars  will  weigh  25.8  pennyweights? 

6.  A  druggist  bought  10  lb.  (Avoir.)  of  oxalic  acid.  How 
many  grains  did  he  buy? 

7.  A  can  contained  20  ounces  (Apoth.)  of  quinine  sulphate. 
How  many  pounds  (Apoth.)  did  it  contain?  How  many 
drams?     How  many  scruples?     How  many  grains? 

8.  A  field  is  20  rods  wide.  How  many  feet  wide  is  it? 
How  many  yards? 


86  GRAMMAR   SCHOOL   ARITHMETIC 

9.    The  perimeter  of  a  square  yard  is  how  many  inches? 

10.  A  100-acre  farm  contains  how  many  square  chains? 

11.  A  cubic  yard  of  earth  is  sometimes  called  a  load.  How 
many  cubic  feet  of  earth  are  there  in  twenty  such  loads? 

12.  Name  the  leap  years  from  1890  to  1920  inclusive. 

13.  What  is  the  exact  number  of  days  from  2  o'clock  p.m., 
Jan.  19,  1904,  to  2  o'clock  p.m.,  April  1,  1904? 

14.  A  gross  of  J-pound  cans  of  baking  powder  will  fill  how 
many  cases,  each  holding  48  cans  ?  How  much  will  the  baking 
powder  weigh? 

15.  How  many  clothespins  are  there  in  a  box  containing  10 
gross  ? 

16.  Four  gallons  of  ammonia  water  will  fill  how  many 
4-ounce  bottles? 

17.  How  many  quires  of  paper  will  a  lady  use  in  writing 
thirty  letters  if  she  uses  two  sheets  of  paper  for  each  letter? 

18.  The  earth  makes  one  complete  rotation  every  24  hours. 
How  many  degrees  does  it  turn  in  1  hour? 

19.  A  wheel  in  a  factory  makes  240  revolutions  per  minute. 
How  many  revolutions  does  it  make  in  one  second  of  time? 
Through  how  many  degrees  does  it  revolve  in  |^  of  a  second? 

20.  A  wheel  has  eight  spokes  that  make  equal  angles  at  the 
center.  How  many  degrees  are  there  in  each  of  the  angles? 
Two  of  these  angles  together  form  what  kind  of  angle  ? 

21.  A  crown  is  an  English  coin  equal  to  five  shillings.  A 
sovereign  is  a  gold  coin  whose  value  is  X 1.  Mr.  Denham  has 
in  his  purse  a  sovereign,  two  crowns,  one  half  crown,  a  shilling, 
and  a  sixpence.  All  the  money  in  the  purse  is  equal  to  how 
many  shillings? 


KEDUCTION   OF  DENOMINATE  NUMBERS  87 

22.  38,476   centimes   are    equal   to   how   many   francs   and 
centimes? 

23.  46  francs  are  equal  to  how  many  centimes? 

24.  86.75  marks  are  equal  to  how  many  pfennigs? 

25.  At  the  rate  of   20  pfennigs  apiece,  how  many  oranges 
can  be  bought  for  4  marks  ? 

26.  Without  a  rule,  draw  a  line  5  feet  long  on  the  black- 
board.    Measure  and  correct  it. 

27.  Estimate^  then  measure: 

a.    The   number  of   feet  from  the  front   door   of  your 

schoolhouse  to  the  sidewalk. 
5.    The  width  of  the  sidewalk. 

c.  The  width  of  the  street. 

d.  The  dimensions  of  the  schoolroom  windows. 

e.  The  dimensions  of  the  schoolroom  doors. 
/.    Other  things  about  the  school. 

203.    1.    Reduce  22  A.  7  sq.  yd.  to  square  feet. 

22 
160 


3520         Number  of  sq.  rd.  in  22  A.  (22  x  160), 


301 


880  (3520  X  1) 
105600  (3520  x  30) 
106480       Number  of  sq.  yd.  in  22  A. 

l_ 

106487       Number  of  sq.  yd.  in  22  A.  7  sq.  yd. 

9 

958383       Number  of  sq.  ft.  in  22  A.  7  sq.  yd. 


GRAMMAR   SCHOOL   ARITHMETIC 
Reduce  392,429  sec.  to  larger  denominations. 


6j9 
6jZI 
24 


39242^  sec. 


654j^  min.  29  sec. 


109  h 


r. 


4  da.  13  hr. 

4  da.  13  hr.  29  sec.   Ans, 


Note.  —  Compound  numbers,  other  than  those  expressing  time,  or  arc 
and  angle  measure,  are  seldom  expressed  in  more  than  two  denominations. 
Extended  reductions  are  rarely  needed. 

In  actual  business,  the  work  of  reduction  is  performed  by 
short  and  direct  processes.  For  example,  surveyors,  in  meas- 
uring land,  use  a  metallic  tape  from  fifty,  to  one  hundred  feet 
in  length,  marked  off  in  feet  and  tenths  of  a  foot,  or  a  chain 
with  links  one  foot  in  length,  marked  in  tenths.  With  this 
they  obtain  the  dimensions  of  a  piece  of  land  in  feet  and  tenths 
of  a  foot,  and  the  area  in  square  feet  and  hundredths  of  a 
square  foot.  The  area  in  square  feet  divided  by  43,560  (the 
number  of  square  feet  in  one  acre)  gives  the  number  of  acres. 

Feet  are  reduced  to  miles  by  dividing  by  5280  instead  of 
dividing  successively  by  the  numbers  in  the  scale  of  linear 
measure. 

Bushels  are  reduced  to  quarts  by  multiplying  directly  by  32. 

In  all  measurements  and  computations,  decimals  are  more 
generally  used  than  formerly,  taking  the  place  of  common 
fractions  and  the  smaller  units  of  denominate  numbers. 

In  the  following  examples,  use  short  and  direct  processes 
where  possible. 

204.     Written 

1.   Reduce  : 

a.   14  wk.  3  da.  to  hours. 

h,   5  T.  7  cwt.  to  pounds. 


REDUCTION  OF  DENOMINATE  NUMBERS  89 

c.  4900  min.  to  higher  denommations. 

d.  7  mi.  to  inches. 

e.  18  bbl.  13  gal.  to  pints. 

/.  193,479  cu.  in.  to  higher  denominations. 

g.  498,342  sec.  to  higher  denominations. 

h.  800,000  oz.  to  tons. 

^.  86,240  pwt.  to  pounds. 

y,  9  oz.  Apoth.  to  grains. 

k,  84,763c?.  to  pounds,  shillings,  and  pence. 

1.  £5  lOs.  lid.  to  pence. 
m,  48°  50'  19' '  to  seconds. 

n.   12  common  years  to  minutes. 

0.   190,113  in.  to  higher  denominations. 

p.   5040  pt.  to  hogsheads. 

q,   3  yr.  7  mo.  21  da.  to  minutes.    (Use  30  da.  for  one  month.) 

r.   4391  da.  to  years  and  days.     (Use  365  days  for  a  year.) 

«.   17  A.  30  sq.  rd.  to  square  feet. 

t.   5  cu.  yd.  to  cubic  inches. 

u,   118,096  sq.  yd.  to  higher  denominations. 

V,   834,769  cu.  in.  to  higher  denominations. 

2.  12,480  in.  are  what  part  of  a  mile  ? 
This  problem  may  be  solved  in  two  ways : 

^'   ell  6  0  =  ^     (There  are  63,360  in.  in  a  mile.) 

3.  2^^  lb.  Troy  =  how  many  grains? 

In  what  other  way  could  this  problem  be  solved? 
(1  lb.  Troy  =  how  manj?^  grains?) 

4.  Change  |  cu.  yd.  to  cubic  inches. 

5.  270  sec.  are  what  part  of  a  day? 


yU  GRAMxMAR  SCHOOL   AKITHMETIC 

6.  Change  yJq  oz.  Apoth.  to  grains. 

7.  108  A.  are  what  part  of  a  square  mile? 

8.  5  rd.  7  ft.  6  in.  are  what  fraction  of  a  mile? 

9.  18  lead  pencils  are  what  part  of  a  gross? 

10.  -^^  cu.  yd.  =how  many  cubic  inches? 

11.  Find  hy  reduction : 

a.  The  number  of  grains  in  1  lb.  Apoth. 

h.  The  number  of  grains  in  a  Troy  pound. 

e.  The  number  of  square  feet  in  1  acre. 

d.  The  number  of  inches  in  a  mile. 

e.  The  number  of  grains  in  1  oz.  Troy. 
/.  The  number  of  grains  in  1  oz.  Apoth. 

12.  Using  the  table  of  equivalents,  page  84,  find : 

a.  The  number  of  cubic  inches  in  one  quart,  liquid  measure, 

b.  The  number  of  cubic  inches  in  one  quart,  dry  measure. 

c.  The  number  of  grains  in  one  ounce,  Apothecaries'  weight. 

d.  The  number  of  grains  in  one  ounce,  Avoirdupois  weight. 

e.  The  number  of  pounds  Troy  that  are  equivalent  to  one 
pound  Avoirdupois. 

13.  Find,  to  the  nearest  thousandth : 

a.  The  number  of  cubic  feet  that  are  equivalent  to  one 
bushel. 

b.  The  number  of  francs  that  are  equivalent  to  one  dollar. 

c.  The  number  of  cents  that  are  equivalent  to  four  marks. 

d.  The  number  of  cents  that  are  equivalent  to  one  shilling. 

e.  The  difference  in  size  between  a  dry  pint  and  a  liquid  pint. 
/.  The  number  of  pounds  Troy  that  are  equivalent  to  10  lb. 

Avoirdupois. 

g.  The  number  of  bushels  that  a  40-gallon  cask  will  hold. 

14.  How  many  5-grain  tablets   can   be   made  from    7J   lb. 
(Avoir.)  of  potassium  chlorate  ? 


ADDITION   OF   COMPOUND   NUMBERS  91 

15.    Make  and  solve  : 

a.  A  problem  that  requires  reduction  descending  in  linear 
measure. 

b.  A  problem  that  requires  reduction  ascending  in  English 
money. 

e.    A  problem,  that  requires  reduction  of  a  denominate  frac- 
tion to  an  integer  of  smaller  denomination. 

d.  A  problem  that  requires   reduction  of   an   integer  to  a 
fraction  of  higher  denomination. 

e,  A  problem  that  requires  reduction  descending  in  square 
measure. 

/.    A  problem  that  requires  reduction   ascending  in  square 
measure. 

ADDITION  AND  SUBTRACTION  OF  COMPOUND  NUMBERS 
205.     Written 

Add  7  lb.  8  oz.,  15  lb.  14  oz.,  23  lb.  15  oz. 
Lb.         Oz. 

15  oz.  +  14  oz.  +  8  oz.  =  37  oz.  =  2  lb.  5  oz. 
2  lb.  +  23  lb.  +  15  lb.  +  7  lb.  =  47  lb. 
47  lb.  5  oz.     Ans. 


1.  59  ft.  8  in.,  47  ft.  11  in.,  9  ft.  9  in. 

2.  63  A.  16  sq.  rd.,  49  A.  53  sq.  rd. 

3.  5  hr.  5  min.  30  sec,  8  hr.  43  min.  47  sec, 

4.  18  gal.  3  qt.,  25  gal.  1  qt.,  16  gal.  2  qt. 

5.  41°  19'  35^  22°  50'  29",  133°  4'  50". 

6.  16  T.  480  lb.,  17  T.  730  lb.,  19  T.  900  lb. 

7.  25  yd.  2  ft.,  6  yd.  1  ft.,  8  yd.  2  ft. 


7 

8 

15 

14 

23 

15 

47 

5 

Add: 

92  GRAMMAR  SCHOOL  ARITHMETIC 

8.  16  pk.  7  qt.,  13  pk.  5  qt.,  12  pk.  6  qt. 

9.  27  cu.  yd.  18  cu.  ft.,  42  cu.  yd.  19  cu.  ft. 

10.  26  yr.  7  mo.  8  da.,  17  yr.  8  mo.  9  da. 

11.  8  lb.  7  oz.,  16  lb.  14  oz.,  19  lb.  10  oz. 

12.  1  bu.  3  pk.,  19  bu.  2  pk.,  5  bu.  1  pk. 

13.  12  wk.  13  da.,  1  wk.  1  da.,  25  wk.  6  da. 

14.  27  hr.  38  min.  21  sec,  25  hr.  47  min.  29  sec. 

15.  25  yr.  200  da.,  27  yr.  321  da.,  28  yr.  179  da. 

206.    Written 

From  18  yr.    7  mo.  14  da. 
take       6  yr.     8  mo.  26  da. 

11  yr.  10  mo.  18  da.     Difference 

7  mo.  14  da.  =  6  mo.  44  da.  „.       ,  ,      ,  ,      .       «x 

- o        o  -tn        1  o  (Why  do  we  make  these  reductions  ?) 

18  yr.  6  mo.  =  17  yr.  18  mo.  v       j  j 

18  yr.  7  mo.  14  da.  =  17  yr.  18  mo.  44  da. 

17  yr.  18  mo.  44  da.  -  6  yr.  8  mo.  26  da.  =  11  yr.  10  mo.  18  da. 

Subtract : 

1.    18  yr.   3  mo.  14  da.  5.    14  yr.  2  mo.  28  da. 

10  yr.   1  mo.   18  da.  1  yr.   9  mo.   12  da. 


2. 

19  yr. 
3  yr. 

2  mo. 
8  mo. 

Ida. 
27  da. 

3. 

29  yr. 
24  yr. 

2  mo. 
8  mo. 

8  da. 
8  da. 

4. 

42  yr. 

28  yr. 

2  mo. 

26  da. 
12  da. 

;.  17  yr. 
3  yr. 

4  mo. 
6  mo. 

8  da. 
7  da. 

'.  29  yr. 

8'yr. 

9  mo. 
5  mo. 

13  da. 

18  da. 

i.   4  yr. 

3  mo. 
9  mo. 

15  da. 

SUBTRACTION  OF  COMPOUND  NUMBERS       93 

9.    How   many   years,  months,  and   days   are   there    from 

May  30,  1907,  to  Dec.  5,  1909  ? 

1909  yr.   12  mo.      5  da.         Note.  —  December  is  the  twelfth  month 

1907  yr.     5  mo.   30  da.     ^"^  ^^^  *^®  ^^^^'     ^^^^*  ^^  da.  for  a 
month. 

Find  the  time  from : 

10.  July  29,  1837,  to  Mar.  26,  1888. 

11.  Aug.  20,  1841,  to  Nov.  15,  1908. 

12.  Dec.  17,  1840,  to  Feb.  18,  1896. 

13.  May  14,  1850,  to  Jan.  12,  1860. 

14.  Oct.  29,  1764,  to  Aug.  23,  1860. 

15.  Jan.  22,  1880,  to  June  15,  1903. 

16.  July  20,  1819,  to  Jan.  2,  1893. 

17.  May  8,  1899,  to  Feb.  12,  1908. 

18.  Feb.  11,  1901,  to  Jan.  31,  1906. 
Subtract : 


19. 

15  hr.  52 

min. 

34 

sec. 

25. 

38  bu. 

Ipk. 

11  hr.  50 

min. 

50 

sec. 

27  bu. 

3pk. 

20. 

222°  41' 

15" 

26. 

210  A. 

,  86  sq. 

rd. 

60°  20' 

35" 

94  A 

.  106  sq. 

rd. 

21. 

43  hr.  44 

min. 

26 

sec. 

27. 

46  1b. 

5  oz. 

30  hr.  24 

min. 

48 

sec. 

25  1b. 

12  oz. 

22. 

47  ft.  6 
33  ft.  10 

in. 
in. 

28. 

98  ft. 
67  ft. 

2  in. 
9  in. 

23. 

57  gal.  1 

qt. 

29. 

52'  13" 

48  gal.  3 

jt. 

45'  28 

ff 

24. 

19°  31' 
6°  41' 

30. 

5  min. 
2  min. 

47  sec. 

48  sec. 

94  GRAMMAR  SCHOOL   ARITHMETIC 

31.  War  was  declared  between  the  United  States  and  Spain 
on  the  25th  day  of  April,  1898,  and  hostilities  ceased  on  the 
13th  day  of  August  in  the  same  year.  How  long  did  the  war 
last  ? 

32.  How  much  time  has  elapsed  since  April  14,  1861  ? 

EXACT  DIFFERENCES  BETWEEN  DATES 

207.  Written 

1.  What  is  the  exact  number  of  days  between  Dec.  16, 1895, 
and  March  12,  1896  ? 

Dec.     15 

Jan.      31  There  are  15  days  in   December  after 

Feb.      29  *1^®  16th.     January  has  31  days,  February 

March  12  ^^  (leap  year),  and  March  12,  making  87 

rr-  ,  .                     days.     Always  count  the  last  day. 

Find  the  exact  time  between: 

2.  June  16,  1886,  and  April  7,  1887. 

3.  Nov.  21,  1898,  and  Dec.  14,  1898. 

4.  Jan.  26,  1907,  and  Dec.  21,  1907. 

5.  July  14,  1898,  and  Aug.  12,  1898. 

6.  Jan.  23,  1897,  and  June  4,  1897. 

7.  Sept.  19,  1899,  and  Feb.  16,  1900. 

8.  Nov.  28,  1905,  and  Oct.  26,  1906. 

MULTIPLICATION  AND   DIVISION  OF  COMPOUND  NUMBERS 

208.  1.    Multiply  5  hr.  21  min.  by  7. 

5  hr.  21  min.  7  x  21  min.  =  147  min.  =  2  hr.  27  min. 

7  7  X  5  hr.       =35  hr. 

1  da.  13  hr.  27  min.  Product    35  hr.  +  2  hr.  =  37  hr.  =  1  da.  13  hr. 


MULTIPLICATION   AND  DIVISION  95 

Multiply : 

2.  6  lb.  7  oz.  Avoir,  by  8 

3.  12.  gal.  1  qt.  by  9 

4.  7  ft.  5  in.  by  6 

5.  12  A.  50  sq.  rd.  by  12 

6.  2  hr.  15  min.  30  sec.  by  15 

7.  7°  40^  18''  by  5 

8.  42  min.  17  sec.  by  15 

9.  3  hr.  19  sec.  by  15 

10.  5  hr.  17  min.  19  sec.  by  15 

11.  18  min.  13  sec.  by  15 

12.  Divide  41°  28'  45"  by  15 

41°  -  15  =  2^  and  11°  rem. 

00  aZ  T^ii   n       '  688' -f- 15  =  45' and  13' rem. 

1  46    55     Quotient         ^3/  _  ^^^t,^    730"  +  45"  =  825". 

825" -^15  =  55". 
Divide  hy  15  and  test  your  work : 

13.  40°  20'  19.    38°  1'  24.    8°  40'  45'' 

14.  17°  18'  15"  20.    7'  30"  25.    59' 

15.  1°  29'  21.    41°  42'  26.    17° 

16.  39'  45"  22.    1°  11'  27.    1°  1'  30" 

17.  27°  30"  23.    40°  2'  30"  28.    11°  19' 

18.  14°  15" 


96 


GRAMMAR   SCHOOL   ARITHMETIC 


MEASUREMENTS 

AREAS   OF  PARALLELOGRAMS 

209.    A  plane  figure  bounded  hy  four  straight  lines  is  a  quadri- 
lateral; e,g. 


Quadrilaterals 

210.    Lines  that  are  the  same  distance  apart  throughout  their 
whole  length  are  parallel  lines;  e.g.- 


112.    A   quadrilateral  whose  opposite  sides  are  parallel  is  a 
parallelogram.     Which  of  the  above  figures  are  parallelograms  ? 

212.  A  parallelogram  that  has  four  right  angles  is  a  rectangle. 
Which  of  the  above  figures  are  rectangles  ? 

213.  Two   lines  that  meet  to  form  a  right  angle  are 
perpendicular  to  each  other. 


214.  The  side  on  which  a  figure  is  supposed  to  rest  is  its  base. 

215.  The  perpendicular  distance  from  the  highest  point  of  a  fig- 
ure to  the  base,  or  to  the  base  extended,  is  its  altitude;  e.g. 

B  c 

c 


AREAS   OF  PARALLELOGRAMS 


97 


216.  Figures  are  read  hy  means  of  letters  placed  at  their 
angles.  Thus,  Fig.  1  is  read,  "Oblong  ABCDr  Fig.  2  is 
read,  "  Triangle  ^5(7."  Read  Fig.  3.  The  base  of  Fig.  2 
is  AQ.     The  altitude  of  Fig.  1  is  i>(7  or  AB. 

217.  The  area  of  a  rectangle  is  the  product  of  its  base  and 
altitude  expressed  in  the  same  denomination. 

Note  1.  —  In  computing  the  area  or  volume  of  a  figure,  the  given  dimen- 
sions, if  expressed  in  different  denominations,  should  first  be  changed  to  the 
same  denomination.     Why  ? 

Note  2. — In  giving  dimensions,  the  sign  (')  is  sometimes  used  to  indi- 
cate feet,  and  the  sign  (")  to  indicate  inches;  e.g.  16' =  16  feet;  9"  =  9 
inches. 


218.    Written 

1.    A  rectangular  field  is  60  rd.  long  and  28  rd.  wide. 


How 


many  acres  does  it  contain  ? 

2.  What  is  the  cost  of  paving  an  alley  570  ft.  long  and  23.7 
ft.  wide,  at  $2.15  per  square  yard? 

3.  The  surveyor  found  my  vacant  lot  to  be  8  rd.  long  and 
67.5  ft.  wide.     What  fraction  of  an  acre  does  it  contain  ? 

4.  Along  a  city  street,  where  the  lots  are  all  12  rd.  deep, 
how  many  feet  wide  must  a  lot  be  to  contain  ^  of  an  acre  ? 

219.    Oral 

1.  In  Fig.  5,  how  does 
the  part  K  compare  with 
the  part  Ml 

2.  The  area  of  the  par- 
allelogram AB  CD  com- 
pares how  with  the  area  of 
the  parallelogram  EFCD"^ 


Fio. 


3.    What  is  the  base  of  each  of  these  parallelograms? 


98 


GRAMMAR   SCHOOL   ARITHMETIC 


4.  What  is  the  altitude  ?   What  is  the  area  ? 

5.  How  is  the  area  of  a  rectangle  found  ? 

6.  If  the  base  of  a  rectangle  is  its  length,  the  altitude  is  what  ? 

7.  If  we  know  the  base  and  altitude  of  a  rectangle,  how  may 
we  find  the  area  ? 

8.  Since  any  parallelogram  may  be  made  into  a  rectangle  of 
the  same  base  and  altitude,  how  may  we  find  the  area  of  a  par- 
allelogram ? 

220.  The  area  of  a  parallelogram  is  equal  to  the  product  of 
its  base  and  altitude  expressed  in  the  same  denomination, 

12  rd. 


16 


15 


Fig.   1 


Fig.  2 


Fig.  8 


221.     Written 

1.  Figure  1  represents  what  part  of  a  square  rod  ? 

2.  Figure  2  represents  what  part  of  an  acre  ? 

3.  Figure  3  represents  what  part  of  a  square  rod  ? 

4.  The  area  of  a  parallelogram  is  52  square  rods.     Its  base  is 
132  ft.     What  is  its  altitude  ? 

5.  The  altitude  of   a  parallelogram  is  37  in.;    its  area  is 
74  sq.  ft.     Find  its  base. 


AREAS  OF  TRIANGLES 


222.    A  plane  figure  hounded  hy  three  straight  lines  is  a  triangle; 


e.g. 


AREAS   OF  TRIANGLES 


99 


223.    Oral 


A       ^s.^ 

A     X 

Fig.  1 


Fig.  2 


Fig. 


Fig.  4 


1.  Figures  1,  2,  3,  and  4  are  what  kind  of  figures  ?     What 
kind  of  figures  are  A  and  B  ? 

2.  In  each  of  the  above  figures  how  does  A  compare  with  B  ? 

3.  In  each  of  the  above  figures,  how  do  the  base  of  the  tri- 
angle and  the  base  of  the  parallelogram  compare  ? 

How  do  the  altitude  of  the  parallelogram  and  of  the  triangle 
compare  ? 


4.    How  is  the  area  of  the  parallelogr: 
angle  ? 

224.  The  area  of  a  triangle  i%  equal  to 
its  base  and  altitude  expressed  in  the  sam 

225.  Oral 
Find  the  areas  of  triangles  having  dime 

Base  Altitude 

1.  7  ft.  4  ft. 

2.  1  yd.  1  yd. 

3.  5  in.  20  in. 

4.  1yd.  1ft. 

5.  80  rd.  20  rd. 

226.  Written 


ound  ?    Of  the  tri- 


Base  ^^^  4  £^^  high,  contains 
6.    1  n 

^-    ^  ^and  4  ft.  high,  contains 

8.  5^ 

9.  1  fand  4  ft.  high,  contains 
10.    640  rd.  1  mi. 


160' 


1.  This  figure  represents  a  plot  of  ground 
inclosed  by  three  streets.  What  part  of  an  acre 
does  it  contain  ? 


100  GRAMMAR   SCHOOL  ARITHMETIC 

^y\  2.    This    figure     represents    a    piece     of 

^^    5^1  \      cement   floor    at    a    railroad   station.     Find 
.^ i     \    its  cost  at  $1.08  a  square  yard. 

3.    In    this    figure,    AB  =  54    in.,    CD  q 

=  18  in.,  j57^=27  in.     Find  the  area  of         /T^^"'^^ 
A  CBF  in  square   yards.  /^ '        ^^""^^^^ 

^r."i --^^ 

MEASUREMENT  OF  RECTANGULAR     \i    ^^ 

SOLIDS  \^^ 

227.  A  solid  hounded  hy  six  rectangles  is 

a  rectangular  solid;  e.g,  a  chalk  box.     Give  other  examples. 

228.  A  solid  hounded  hy  six  squares  is  a  cube.     Define  cubic 
ineh^  cubic  foot  ^  and  cubic  yard. 

/^ II 

^ —    229.    The  contents  or  volume  of  a  rec- 

FlG.     1  .  .  7  .  . 

tangular  solid  is  the  number  of  cubic  units 
221.     Written  ^^^^  -^^^^  ^^^  £^^^  ^^^  ^^  ^^  ^^^.^^  ^^  contains, 

1.  Figure  1  represei^,^^^  ^^  equal   to  the  product  of  its  three 

2.  Figure  2  vQ'^vQ^Qidimensions. 

3.  Figure  3  represer    Figure  A  may  represent  5x4x1  cu. 

4.  The  area  of  a  parin.,  cu.  ft.,  or  cu.  yd. 

132  ft.     What  is  its  ali    Figure  B  may  represent  5x4x3  cu. 

5.  The  altitude  of  in.,  cu.  ft.,  or  cu.  yd. 
74  sq.  ft.     Find  its  b    ^3^^    ^^^^ 

.  „     1.    Explain    how   we   determine    that 
there  are  27  cu.  ft.  in  1  cu.  yd. 

2.  Explain  how  we  determine  that  there  are  1728  cu.  in.  in 
1  cu.  ft. 

3.  A  candy  box  is  6  in.  square  and  2  in.  deep.     What  is   its 
volume  ? 


MEASUREMENTS:   THE   CORD 


101 


4.  A  rectangular  piece  of  wood  4  in.  square  and  10  in.  long 
contains  how  many  cubic  inches  ? 

5.  A  piece  of  timber  10  in.  square  and  1  yd.  long  contains 
how  many  cubic  inches?  It  must  be  how  long  to  contain 
1200  cu.  in.?  1350  cu.  in.  ?  1  cu.  ft.? 

6.  A  box  is  10  in.  wide  and  10  in.  deep,  inside  measure. 
What  must  be  its  length  in  order  that  it  may  hold  1  bushel  ? 

7.  The  bottom  of  a  rectangular  tin  can  measures  5J  in.  by 
6  in.     How  deep  must  it  be  to  hold  1  gallon  ? 


231.  Oral  ^  ^^^^ 

1.  A  pile  of  4-foot  wood,  8  ft.  long  and  4  ft.  high,  contains 
how  many  cubic  feet  ? 

2.  A  pile  of  2-foot  wood,  8  ft.  long  and  4  ft.  high,  contains 
how  many  cubic  feet? 

3.  A  pile  of  1-foot  wood,  8  ft.  long  and  4  ft.  high,  contains 
how  many  cubic  feet  ? 

4.  If  wood  is  cut  into  sticks  that  are  1  ft.  6  in.  long,  how 
many  cubic  feet  are  there  in  a  pile  8  ft.  long  and  4  ft.  high  ? 

232.  Originally,  a  cord  of  wood  consisted  of  128  cu.  ft.,  or 
the  equivalent  of  a  pile  of  4-foot  wood,  8  ft.  long  and  4  ft. 
high.      It  is  a  growing  custom,  however,  to  consider  as  a  cord 


102  GRAMMAR   SCHOOL   ARITHMETIC 

any  quantity  of  wood  that  is  equivalent  to  a  pile  8  ft.  long  and 
4  ft.  high,  whatever  may  be  the  length  of  the  sticks.  In  some 
states  the  law  specifies  what  shall  be  understood  as  a  cord  of 
wood. 

To  find  the  number  of  cords  in  a  pile  of  wood^  find  the  product 
of  its  three  dimensions  expressed  in  feet^  and  divide  the  product 
by  128. 

233.  Written 

1.  Find  the  number  of  cords  in  a  pile  of  4-foot  wood  : 

a.  36  ft.  long  and  6  ft.  high. 

b.  20  ft.  long  and  5  ft.  high. 

c.  50  ft.  long  and  8  ft.  high. 

d.  100  ft.  long  and  6  ft.  high. 

e.  8  ft.  long  and  7^  ft.  high. 

2.  What  must  be  the  length  of  the  sticks  in  a  pile  of  wood 
4  ft.  high  and  32  ft.  long  in  order  that  it  may  contain  : 

a.  512  cu.  ft.  ?  c,    256  cu.  ft.  ?  e.    192  cu.  ft.  ?. 

b,  384cu.  ft.  ?  d,    128cu.  ft.  ?  /.    160  cu.  ft.  ? 

3.  What  must  be  the  length  of  a  pile  of  4-foot  w^ood  in  order 
that  a  pile  4  ft.  high  may  contain  5 J  cords  ? 

4.  What  must  be  the  height  of  a  pile  of  4-foot  wood  in  order 
that  a  pile  20  ft.  long  may  contain  : 

a.    2^  cords?  b.    4  cords?  c.    IJ  cords? 

BUILDING   WALLS 

234.  There  are  no  universal  rules  for. the  measurement  of 
masonry.  Some  masons  measure  around  the  outside  of  a  cellar 
wall  to  determine  its  dimensions,  while  others  make  allowance 
for  the  corners.  The  method  ojf  measurement  should  be  speci- 
fied in  the  contract  in  every  case. 


MEASUREMENTS:    BUILDING  WALLS  103 

Quantities  of  uncut  stone  are  bought  by  the  cord,  and  usually 
99  cu.  ft.  are  taken  for  a  cord. 

From  21  to  23  bricks  8''  x  4'^  x  ^"  are  estimated  to  make  a 
cubic  foot  of  brick  wall. 

Some  masons  estimate  the  number  of  bricks  required  for  a 
wall  by  multiplying  the  number  of  square  feet  in  one  side  of 
the  wall  by  7,  when  the  wall  is  one  brick  thick,  by  14  when  it 
is  two  bricks  thick,  and  by  21,  when  it  is  three  bricks  thick, 
allowing  for  all  openings. 

A  perch  of  stone  or  masonry  is  24|  cu.  ft. 

Concrete  w^alls  are  estimated  by  the  cubic  yard,  and  the 
methods  of  measurement  vary.  Foundation  walls  are  generally 
measured  without  regard  to  openings.  When  there  are  many 
openings  some  contractors  allow  one  half  for  openings,  and 
some  make  full  allowance. 

235.    Written 

1.  A  retaining  wall  is  220  ft.  long  and  8  ft.  high.  It  has  an 
average  thickness  of  3  ft.  Find  the  cost  of  the  stone  used,  at 
$5.40  per  cord,  a  cord  of  stone  making  99  cu.  ft.  of  wall. 

2.  Find  the  cost  of  the  brick  for  a  wall  120  ft.  long,  12^  in. 
thick,  and  40  ft.  high,  at  $6.50  per  M.,  estimating  21  bricks  for 
a  cubic  foot,  and  making  no  allowance  for  openings. 

3.  Find  the  cost,  at  $9.50  per  M.,  of  a  brick  veneer  4  in. 
thick  on  the  outside  of  a  house  measuring  45  ft.  by  30  ft.  and 
20  ft.  high,  making  an  allowance  of  200  sq.  ft.  for  doors  and 
windows,  and  allowing  7  bricks  for  a  square  foot  of  surface. 

4.  The  walls  of  a  rectangular  cellar  87  ft.  6  in.  by  45  ft. 
(outside  measure),  and  9  ft.  deep,  are  18^'  thick.  Find  the  cost 
of  the  stone  at  $6.30  per  cord,  estimating  a  cord  of  stone  to 
make  100  cu.  ft.  of  wall,  and  deducting  18  cu.  ft.  for  corners. 


104 


GRAMMAR  SCHOOL  ARITHMETIC 


5.  Find  the  cost  of  24  concrete  pier  foundations  each  21 
in.  square  and  5  ft.  deep,  at  15.00  per  cubic  yard. 

6.  A  garden  wall  55  ft.  long,  6  ft.  high,  and  18  in.  thick 
cost  how  much  at  $4.20  a  perch? 

7.  A  contractor  built  a  concrete  cellar  wall  54  ft.  by  30  ft. 
(outside  measure),  8  ft.  high,  and  IJ  ft.  thick,  receiving  $5  per 
cubic  yard.  He  used  75  barrels  of  cement  costing  $1.80  a 
barrel,  12  loads  of  sand  costing  $1.25  a  load,  and  50  cu.  yd.  of 
crushed  stone  at  $1.30  per  cubic  yard.  How  much  had  he  left 
for  labor  and  profit  ? 

FLOOR  COVERING 

236.  A  yard  of  carpet  or  mat- 
ting is  a  i/ard  of  the  length  of  the 
piece,  as  it  is  unrolled,  regardless 
of  its  width. 

The  exact  number  of  yards  of 
material  to  be  purchased  for  the 
covering  of  any  given  floor  is 
difficult  to  determine,  because  of 
the  waste  in  fitting,  and  in  matching  figures. 

237.    Written 

1.  If  this  floor  is  covered  with  carpet  |  yd.  wide,  how  many 
strips,  running  lengthwise,  must  be  pur- 
chased ? 

Note.  —  When  a  part  of  the  width  of  a 
strip  is  needed,  a  whole  strip  must  generally 
be  purchased. 

2.  How  many  yards  of  carpet  must  be 
purchased  for  this  floor,  allowing  1  yd.  for  waste  in  matching  ? 

3.  A  room  is  20'  x  18'. 


26' 


MEASUREMENTS :    PLASTERING 


105 


a.  Find  the  cost  of  carpeting  the  floor  with  Brussals  carpet, 
27  in.  wide,  at  $1.25  per  yard,  adding  8  cents  a  yard  for  mak- 
ing and  laying,  and  allowing  IJ  yd.  for  waste  in  matching 
figures. 

h.  Find  the  cost  of  covering  the  floor  with  matting  one  yard 
wide,  at  60^  a  yard,  adding  7^  a  yard  for  laying  and  allowing 
nothing  for  waste. 

4.  Find  the  cost  of  carpeting  a  floor  16'  6^'  x  14'  with  ingrain 
carpet,  1  yd.  wide,  at  f. 75  a  yard,  allowing  \\  yd.  for  waste  in 
matching,  and  covering  the  floor  first  with  carpet  paper  at  4^ 
a  square  yard. 

5.  A  room  is  45  ft.  by  25  ft.  How  many  yards  of  carpet 
I  yd.  wide  are  needed  to  cover  the  floor,  running  the  strips  so 
as  not  to  divide  a  strip  ? 

6.  An  office  floor  18'  x  27'  is  covereti  with  inlaid  linoleum 
\\  yd.  wide.  Find  its  cost  at  $1.40  per  square  yard,  allowing 
one  square  yard  for  matching. 

7.  Measure  your  schoolroom  and  compute  the  cost  of  carpet- 
ing a  room  of  the  same  size  with  velvet  carpet  |  yd.  wide  at 
$1.30  per  yard. 


PLASTERING 

238.    The  cost  of  plastering  is  estimated  by  the  square  yard. 

Some  contractors  deduct  the  entire  sur- 
face of  doors,  windows,  and  other  open- 
ings, and  some  deduct  only  one  half  of 
such  surfaces. 


12' 
End 
Wall 


16' 
Side 
Wall 


12' 
End 
Wall 


16' 


Ceiling 


16' 

Side 
Wall 


106  GRAMMAR  SCHOOL  ARITHMETIC 

239.     Written 

1.  Tlie  cut  on  page  105  represents  the  four  walls  and  ceiling 
of  a  room. 

a.  What  is  the  entire  length  of  the  end  and  side  walls  ? 
What  is  the  height  ? 

h.    How  many  square  feet  are  there  in  all  the  walls  ? 

c.  How  many  square  feet  are  there  in  the  ceiling  ? 

d.  How  many  square  feet  are  there  in  the  walls  and  ceiling 
together? 

e.  How  many  square  yards  are  there  in  all  ? 

f.  What  will  it  cost  to  lath  and  plaster  this  room  at  35  cents 
a  square  yard,  taking  out  5|  square  yards  for  openings  ? 

2.  A  schoolroom  is  40  ft.  square  and  14  ft.  high.  The 
wainscoting  is  3  ft.  8  in.  high. 

a.  Find  the  cost  of  lathing  and  plastering  the  four  walls  at 
38^  per  square  yard,  making  full  allowance  for  10  windows 
4  ft.  X  7 J  ft.,  and  no  allowance  for  doors. 

b.  Find  the  cost  of  a  steel  ceiling  for  this  room  at  9^  per 
square  foot. 

3.  Find  the  cost  of  lathing  and  plastering  the  walls  and 
ceiling  of  a  room  19  ft.  by  36  ft.  and  12  ft.  high  at  36  ^  per 
square  yard,  making  one  half  allowance  for  3  doors  each  3  ft. 
8  in.  by  8  ft.,  and  six  windows  each  4  ft.  by  7 J  ft. 

-    4.    Measure  the  plastered  parts  of  your    schoolroom  to  the 
nearest  half  of  a  foot. 

a.  Find  the  cost  of  metal  laths  at  18/  a  square  yard,  suf- 
ficient for  this  room,  making  full  allowance  for  doors  and 
windows. 

b.  If  a  contractor  received  60^  per  square  yard  for  lathing 
and  plastering  the  room,  using  the  answer  to  question  a  for  the 


MEASUREMENTS:    WALL   COVERINGS  107 

cost  of  the  laths,  find  what  the  labor  and  the  remaining  ma- 
terials cost. 

c.  Find  the  cost  of  wood  laths  sufficient  for  this  room  at 
$5.75  per  M.,  estimating  a  bundle  of  50  laths  to  cover  2| 
square  yards. 

WALL  COVERINGS 

240.  A  roll  of  figured  wall  paper  is  usually  8  yards  long  and 
J  yard  wide.  How  many  square  yards  of  paper  does  it  con- 
tain ? 

Ingrain  paper  is  30  inches  wide. 

Paper  hangers  generally  estimate  that  a  roll  of  paper  will 
cover  from  30  to  34  square  feet  of  wall,  after  allowing  for 
waste. 

Woven  wall  coverings  are  sold  by  the  square  yard. 

241.  Written 

1.  A  room  22'  x  16J'  and  10  feet  high  was  papered  entirely 
with  figured  wall  paper  costing  30  cents  a  roll. 

A  molding  costing  5  cents  a  lineal  foot  extended  around  the 
top  of  the  wall.  Two  men  did  the  work  in  one  day  and  re- 
ceived i3.75  each. 

a.  Find  the  cost  of  decorating  the  room,  allowing  for  one 
window  4|  ft.  by  6  ft.,  two  windows  3  ft.  4  in.  by  6  ft.,  and 
a  baseboard  12  in.  high,  and  estimating  a  roll  of  paper  to  cover 
32  square  feet  of  surface. 

h.  How  much  would  the  ceiling  have  cost  if,  instead  of  being 
papered,  it  had  been  covered  with  prepared  muslin  costing  20 
cents  a  square  yard  and  tinted  with  material  costing  45  cents 
and  requiring  1  day's  labor  for  two  men  ? 

2.  Find  the  cost  of  decorating  a  dining  room  14'  x  18'  and 
9^  ft.  high,  as  follows:  the  side  walls  covered  with  plain  bur- 
lap at  25  cents  a  square  yard ;  the  ceiling  covered  with  paper  at 


108  GRAMMAR  SCHOOL  ARITHMETIC 

10  cents  a  roll,  a  roll  covering  30  square  feet ;  picture  mold- 
ing and  plate  rail  costing  $12;  water  colors,  glue,  flour,  etc., 
65  cents;  allowance  made  for  100  square  feet  of  openings; 
labor,  2|-  days  for  two  men  at  $  3|  per  day  for  each  man. 

3.  a.  Select  a  room  in  your  own  home.  Find  the  cost  of 
decorating  it  as  your  mother  would  like  to  have  it  done.  Ask 
her  what  she  would  like  to  have  put  on  the  walls;  then  you 
make  the  measurements,  compute  the  amount  of  material  and 
labor,  and  the  cost. 

b.  Decorate  in  the  same  way  a  room  15  ft.  long,  12  ft.  wide, 
and  9  ft.  high. 

LUMBER  MEASURE 

242.  A  piece  of  wood  1  ft.  long,  1  ft.  wide,  and  1  in.  thick 
is  a  board  foot  (bd.  ft.). 

To  THE  Teacher.  —  As  material  for  this  lesson,  a  real  board  foot  —  a 
piece  of  board  exactly  1  ft.  long,  1  ft.  wide,  and  1  in.  thick  —  should  be  pro- 
vided. Refer  to  it  in  obtaining  answers  to  the  oral  questions  below  and 
whenever  pupils  seem  to  answer  wide  of  the  mark  in  this  subject.  This  is 
very  important. 

243.  Oral 

1.  A  board  foot  contains  how  many  cubic  inches  ? 

2.  How  many  board  feet  piled  one  upon  another  would 
make  a  cubic  foot  of  lumber  ? 

3.  A  board  foot  is  what  part  of  a  cubic  foot  ? 

4.  A  piece  of  lumber  10'  x  1'  x  1"  contains  how  many 
board  feet  ? 

5.  If  the  lumber  in  Question  4  were  2  in.  thick,  how  many 
board  feet  would  it  contain  ?     If  it  were  5  in.  thick  ?  7  in.  thick  ? 


LUMBER  MEASURE  109 

6.  The  floor  of  a  room  10  feet  square  is  1  in.  thick.  How 
many  feet  of  boards  does  it  contain  ? 

7.  A  bridge  15  ft.  long  and  10  ft.  wide  is  floored  with 
3-inch  plank.     How  many  feet  of  plank  are  there  in  the  floor  ? 

8.  A  board  8  ft.  long,  3  in.  wide,  and  1  in.  thick  contains 
how  many  board  feet  ? 

9.  A  timber  is  30  ft.  long  and  one  foot  square.  Walk  as 
far  as  this  timber  would  reach.  Show  with  your  hands  its 
width  and  thickness.  How  many  cubic  feet  of  lumber  does  it 
contain  ?     How  many  board  feet  ? 

10.  A  board  16  ft.  long  and  1  in.  thick  must  be  how  wide  to 
contain  8  board  feet  ? 

244.  We  may  find  the  number  of  board  feet  in  a  piece  of 
lumber  by  multiplying  the  number  of  cubic  feet  by  12.  The 
rule  commonly  used  by  dealers  and  mechanics  gives  the  same 
result,  and  is  stated  as  follows  : 

To  find  the  number  of  hoard  feet  in  any  piece  of  lumber^  mul- 
tiply together  its  three  dimensions,  two  of  them  expressed  in  feet 
and  the  other  in  inches. 

The  cost  of  25  planks  each  16  ft.  long,  11  in.  wide,  and  3  in„ 
thick,  at  i  28  per  thousand  feet,  may  be  found  thus  : 

Lumber  that  is  less  than  1  in.  thick  is  counted  as  1  in.  thick 
in  measuring. 


no 


GRAMMAR  SCHOOL  ARITHMETIC 


KEY   TO   ILLUSTRATION 
a.   Outside  studding 
h.   Rafters 

c.  Plates 

d.  Ceiling  joists 
de.   Second  floor  joists 

def.   First  floor  joists 


g.   Girder 
h.    Sills 
i.    Sheathing 
/.    Partition  studs 
k.   Partition  heads 
/.  Piers 


m.   Foundation 
245.     Written 

1.    Find  the  cost  of  the  following  bill  of  lumber  : 
Note. — M.  stands  for  thousand  feet. 

4  sills  6"  X  10"  X  16',  $27  per  Me 

2  sills  6"  X  10"  X  18',  127  per  M. 

1  girder  8"  x  10"  x  18',  |27  per  M. 

26  rafters  2"  x     6"  x  14',  $27  per  M. 

60  pieces  of  studding  2"  x    4"  x  16',  $  27  per  M, 


LUMBER  MEASURE 


111 


Flooring  for  three  floors  18'  x  30'  x  |",  138  per  M. 
2000  feet  of  sheathing,  1 30  per  M. 
200  feet  of  casings,  |45  per  M. 

2.  What  is  the  cost  of  10  joists,  each  16  ft.  long,  10  in.  wide, 
and  3  in.  thick,  at  $  26  per  M.  ? 

3.  Find  the  cost  of  a  stick  of  timber  8  in.  square,  and  30  ft. 
long,  at  $18  per  M. 

4.  What  is  the  cost  of  8  sticks  of  timber  each  36  ft.  long, 
10  in.  wide,  8  in.  thick,  at  |18  per  M.? 

5.  I  need  213  planks  4  ft.  8  in.  long,  1  ft.  wide,  and  IJ  in. 
thick,  to  build  a  sidewalk.  How  much  will  they  cost  at  1 25 
a  thousand  ? 

6.  A  builder  bought  425  half-inch  boards  16  ft.  long  and  2-|- 
in.  wide.     How  many  feet  of  lumber  did  he  buy  ? 

7.  How  many  board  feet  are  there  in  24  joists  16'  x  14'^  x  3^'? 

8.  How  many  feet  of  2-inch  plank  will  cover  a  barn  floor 
20  ft.  wide  and  60  ft.  long  ? 

9.  a.  These  figures  represent  one  end  and 
one  side  of  a  building  covered  with  clap- 
boards I  of  an  inch  thick  that  cost  $36  per 
M.  Allowing  ^  of  all  the  lumber  purchased, 
for  waste  in  cutting  and  overlapping,  how 
much  did  the  clapboards  for  this  building 
cost? 

Hint.  —  If  there  were  no  waste,  how  much  lumber 
would  be  needed?  This  is  what  part  of  the  lumber 
purchased,  when  J  of  the  lumber  purchased  is  wasted  ? 

h. 
X  6"  and  25  rafters  are  used 
on  each  side  of  the  roof.     How 
much  did  they  cost  at  $27  per  M.? 


48' 


The  rafters   are   20' x  2^' 


112 


GRAMMAR   SCHOOL   ARITHMETIC 


c.  The  roof-boards  are  nailed  to  the  rafters,  with  spaces  be- 
tween the  boards,  so  that  only  |^  of  the  surface  of  the  roof  is 
covered  with  boards.  What  is  the  cost  of  the  roof-boards  for 
the  roof  of  this  building  at  $24  per  M.? 


ESTIMATING  SHINGLES 

246.  Oral 

1.  In  measuring  shingles,  the  average  width  of  the  shingles 
is  supposed  to  be  4  inches.     The  length  varies,  but  they  are 

always  laid  so  that  more  than 
two  thirds  of  the  shingle  is 
covered  by  the  courses  of  shin- 
gles above.  If  they  are  laid 
so  that  5  inches  of  the  length 
are  exposed  to  the  weather, 
a  shingle  4  inches  wide  will  really  make  how  many  square 
inches  of  roof?  1000  shingles  will  make  how  many  square 
inches  of  roof  ? 

2.  When  shingles  are  laid  6  inches  to  the  weather,  each 
shingle  will  make  how  many  square  inches  of  roof  ?  How 
many  shingles  will  make  one  square  foot  of  roof  ?  How  many 
shingles  are  required  for  one  square  (100  square  feet)  of  roof  ? 

3.  When  shingles  are  laid  4|^  inches  to  the  weather,  one 
shingle  makes  how  many  square  inches  of  roof  ?  How  many 
shingles  will  make  one  square  foot  of  roof?  One  square  of 
roof? 

247.  Written 

1.  a.  How  many  shingles  laid  6  inches  to  the  weather  are 
required  for  one  square  foot  of  roof  ?  h.  For  one  square  of 
roof  ?  c.  For  a  roof  20  ft.  long,  each  slanting  side  of  which 
measures  9  ft.  in  width? 


MEASUREMENTS:    VOLUME   AND   CAPACITY  113 

2.  Find  the  cost  of  the  shingles  for 
this  roof  at  |4.50  per  M.  (1000  shingles), 
estimating  that  675  shingles  will  make 
one  square  of  roof. 

3.  Find  the  cost  of  the  shingles  for  a  roof  36  ft.  long,  each 
slanting  side  19  ft.  wide;  the  shingles  being  laid  so  that 
seven  shingles  make  one  square  foot  of  roof,  and  costing  |5.20 
per  M. 

4.  A  shed  roof  18^  x  40'  slants  only  one  way.  Find  the  cost 
of  the  shingles  required  for  it  at  14.80  per  M.,  the  shingles 
being  laid  so  that  1^  shingles  make  one  square  foot  of  roof. 

5.  Find  the  cost  of  the  shingles  at  i  6  per  M.  to  cover  15 
squares  of  roof,  the  shingles  being  laid  6  inches  to  the  weather. 

6.  The  shingles  for  a  roof  cost  $68.25.  Each  side  of  the 
roof  measured  25  ft.  by  35  ft.,  and  the  shingles  were  laid  so  that 
6|-  shingles  made  a  square  foot  of  roof.     Find  the  price  per  M. 

VOLUME  AND  CAPACITY 
248.    Oral 

1.  In  what  denominations  are  measures  of  volume  expressed? 
Measures  of  capacity  ? 

2.  One  gallon  is  equal  to  how  many  cubic  inches  ? 

3.  One  bushel  is  equal  to  how  many  cubic  inches  ? 

4.  When  the  volume,  in  cubic  inches,  of  a  tank,  cistern,  or 
cask,  is  known,  how  may  its  capacity  in  gallons  be  found  ? 

5.  When  the  volume,  in  cubic  inches,  of  a  box,  bin,  or  barrel, 
is  known,  how  may  its  capacity  in  bushels  be  found  ? 

6.  What  are  the  prime  factors  of  231  ? 

7.  How  may  we  find  the  capacity  of  a  bin  in  bushels,  when 
we  know  its  dimensions  in  inches  ?     In  feet  ? 


114  GRAMMAR  SCHOOL   ARITHMETIC 

8.  When  we  know  the  capacity  of  a  bin  in  bushels,  how  may 
we  find  its  volume  in  cubic  inches  ?     In  cubic  feet  ? 

9.  A  rectangular  tin  can  7  in.  by  3  in.  by  11  in.  will  hold 
how  many  liquid  quarts  ? 

10.  The  volume  of  a  bin  is  215,042  cubic  inches.     How  many 
bushels  will  it  hold  ? 

11.  The  volume  of  a  keg  is  2310  cubic  inches.     How  many 
gallons  will  it  hold  ? 

12.  What  is  the  volume  of  a  cask  that  holds  100  gallons  ? 

13.  The  volume  of  a  rectangular  solid  is  the  product  of  how 
many  dimensions  ? 

14.  The  dimensions  are  what  of  the  volume  ? 

15.  When  three  factors  are  known,  how  may  their  product 
be  obtained  ? 

16.  When  a  product  and  two  of  its  three  factors  are  known, 
how  may  the  other  factor  be  found  ? 

17.  When  the  dimensions  of  a  rectangular  solid  are  known, 
how  may  its  volume  be  found  ? 

18.  When  the  volume  and  two  dimensions  of  a  rectangular 
solid  are  known,  how  may  the  other  dimension  be  found  ? 

19.  A  box  6  in.  by  8  in.  must  be  how  deep  to  contain  96 
cu.  in.  ? 

20.  A  box  5''  X  5'^  X  ?  contains  100  cu.  in. 

21.  A  box    ?  X  11'^  X  3''  holds  231  cu.  in. 

22.  A  box  7^'  X  ?  X  3'^  holds  231  cu.  in. 

23.  A  rectangular  tin  box  is  11  inches  long  and  7  inches  wide 
and  holds  a  gallon.     How  deep  is  it  ? 


MEASUREMENTS :    VOLUME   AND   CAPACITY  115 

249.    Written 

1.  Find  in  gallons  the  capacity  of  a  cistern  11  ft.  square 
and  6  ft.  deep. 

2.  Find  to  the  nearest  hundredth  («)  the  number  of  gallons 
that  are  equivalent  to  one  cubic  foot ;  (b')  the  number  of 
bushels  that  are  equivalent  to  one  cubic  foot. 

3.  Find  to  the  nearest  hundredth  the  number  of  gallons  that 
are  equivalent  to  one  bushel. 

4.  A  box  car  is  33  ft.  long,  8  ft.  8  in.  wide,  and  7  ft.  6  in.  high, 
inside  measure.  It  is  strong  enough  to  carry  30  tons.  A 
bushel  of  corn  weighs  60  pounds. 

a.    How  many  bushels  of  corn  can  the  car  carry  ? 
h.    How  many  cubic  feet  (to  the  nearest  tenth  of  a  cubic 
foot)  will  the  load  occupy  ? 

c.  How  many  cubic  feet  of  space  will  be  left  unoccupied? 

d.  A  bushel  of  oats  weighs  32  pounds.  How  much  space, 
to  the  nearest  tenth  of  a  cubic  foot,  will  be  left  when  the  car 
contains  20  tons  of  oats  ?     (Allow  IJ  cu.  ft.  for  a  bushel.) 

5.  A  cellar  35  ft.  long  and  21  ft.  wide  was  flooded,  during 
a  storm,  to  a  depth  of  3  ft.  8  in.  What  was  the  cost  of  pump- 
ing out  the  water  at  $.03  a  barrel ? 

6.  A  watering  trough  in  the  form  of  a  rectangular  box  is 
11  ft.  long,  18  in.  wide,  and  14  in.  deep.  How  many  barrels 
of  water  will  it  hold?     (Result  correct  to  hundredths.) 

7.  A  farmer,  having  ten  44-gallon  casks,  used  them  for 
storing  wheat.  How  many  bushels  of  wheat  could  he  store  in 
them?     (Result  correct  to  hundredths.) 

8.  I  have  in  my  attic  a  rectangular  copper  water  tank  14  ft. 
by  9  ft.,  into  which  the  rain-water  from  the  roof  is  carried. 
During  a  shower,  the  tank  was  filled  to  a  depth  of  11  inches. 
How  many  barrels  of  water  ran  into  it  ? 


116  GRAMMAR  SCHOOL  ARITHMETIC 

9.  A  reservoir  from  which  a  city  is  supplied  with  water  has 
a  surface  of  35  acres.  If  no  water  ran  into  it,  the  surface  of 
the  water  would  be  lowered  5  inches  a  day  by  the  pipes  that 
supply  the  city.     How  many  gallons  are  used  daily  ? 

10.  A  teamster  wanted  to  know  how  many  gallons  of  water 
he  could  carry  in  his  watering-pail.  He  had  no  measure  except 
a  foot  rule.  He  measured  a  feed  box  and  found  the  inside 
dimensions  to  be  :  length  2  ft.  9  in.,  width  1  ft.  9  in.,  depth  1  ft. 
He  filled  the  pail  with  oats  and  emptied  them  into  the  box, 
repeating  the  process  till  the  box  was  full.  The  box  held  twelve 
pails  of  oats.  Find,  (a),  the  volume  of  the  box  in  cubic  inches, 
(5),  the  volume  of  the  pail,  (c),  the  capacity  of  the  pail  in  gallons. 

11.  Some  boys  found  a  bowlder,  and  guessed  the  number  of 
cubic  inches  of  stone  that  it  contained.  To  find  which  was  the 
best  guesser,  they  filled  a  large  pail  with  water  and  set  it  in  an 
empty  washtub.  Then  they  placed  the  bowlder  in  the  pail  of 
water  so  that  the  bowlder  was  entirely  submerged,  and  found 
that  5  qt.  1  pt.  of  water  had  run  over  into  the  washtub.  The 
nearest  guess  was  350  cu.  in.  Was  it  too  large,  or  too  small, 
and  how  much  ? 

12.  A  cubic  foot  of  water  weighs  62J  lb.  What  is  the 
weight  of  a  gallon  of  water?     (Correct  to  3  dec.  places.) 

13.  The  water  displaced  by  a  floating  body  weighs  the  same 
as  the  floating  body.  A  log  containing  20  cu.  ft.  of  wood,  float- 
ing in  a  stream,  was  three  fourths  under  water. 

a.    How  many  gallons  of  water  did  it  displace  ?  (2  dec.  places.) 
h.    What  was  its  weight  ? 

14.  My  house  covers  a  surface  equivalent  to  a  rectangle 
20'  X  40^  During  a  rain  storm,  water  fell  to  an  average  depth 
of  .8  of  an  inch,  according  to  the  record  at  our  weather  station. 
How  many  barrels  of  water  fell  on  my  roof  ?     (2  dec.  places.) 


REVIEW  AND  PRACTICE  117 

15.  A  farmer's  wagon  box  was  3  ft.  4  in.  wide,  16  ft.  6  in. 
long,  and  20  in.  deep.  Find,  to  the  nearest  tenth,  the  number 
of  bushels  that  it  holds. 

16.  A  wagon  box  12  ft.  long  and  3  ft.  6  in.  wide  holds  40 
bushels.      Find  its  depth  to  the  nearest  tenth  of  an  inch. 

17.  A  fruit  grower  made  some  bushel  crates  that  were  2  ft. 
long  and  1  ft.  deep.  Find  their  width  to  the  nearest  tenth  of 
an  inch. 

18.  An  aquarium  is  7  ft.  long  and  22  in.  wide. 

a.  When  it  contains  40  gallons  of  water,  how  deep  is  the 
water  ? 

h.  How  deep  is  the  water  when  it  contains  one  hogshead  of 
water  ? 

c.  When  the  water  is  two  feet  deep,  how  many  gallons  does 
the  aquarium  contain? 

d.  When  the  water  is  8.64  in.  deep,  how  many  pounds  of 
water  are  there  in  the  aquarium  ?     (See  Question  12.) 

19.  How  many  barrels  of  water  will  a  rectangular  cistern 
6'  X  5'  X  41^  hold  ? 

REVIEW  AND  PRACTICE 
250.     Oral 

1.  Read  CLI ;  MCMIX ;  CDLXXXVIII ;  CCXVI. 

2.  Read  10.0010  ;  100.00100;  101.00001;  101.100. 

3.  Give  results  rapidly  :    . 

38  +  45;  98-79;  98  +  34;  78x99;  60x80;  1.047x100; 
96x25;  315X.33J;  12x25;  1300-^25;  48x125; 
428.3  -- 1000 ;    125  x  2000 ;    360,000  -v-  400. 

4.  What  is  the  smallest  number  that  exactly  contains  2,  3, 
4,  6,  and  8? 


118  GRAMMAR  SCHOOL   ARITHMETIC 

5.  What  is  the  largest  number  that  will  exactly  divide  45, 
60,  and  75? 

6.  27  is  a  power  of  what  number? 

7.  Name  four  powers  of  10. 

8.  How  is  the  value  of  a  figure  affected  by  moving  it  three 
places  to  the  left? 

9.  Of  what  number  are  5,  2,  and  13  the  prime  factors? 

10.  The  product  of  two  or  more  numbers  is  found  by  what 
operation  ? 

11.  One  of  the  two  factors  of  a  number  is  found  by  what 
operation,  when  the  product  and  the  other  factor  are  known? 

12.  Describe  two  tests  for  examples  in  subtraction. 

13.  The  product  of  three  factors  contains  five  decimal 
places.  One  of  the  factors  has  three  decimal  places  and 
another  two.     How  many  decimal  places  has  the  third  factor? 

14f    Name  four  signs  of  aggregation. 

15.  3  +  18  ^  6  -  2  X  3  =  ? 

16.  How  can  you  tell  whether  a  number  is  divisible  by 

(a)  2,  (5)  3,  (0  4,  id)  5,  (6)  6,  (/)  8,  (^)  10,  (K)  9? 

17.  The  sum  of  the  digits  in  a  number  is  27.  What  num- 
bers will  divide  it? 

18.  The  sum  of  the  digits  in  a  number  is  18  and  the  figure 
in  units'  place  is  8.     What  numbers  will  divide  it? 

19.  The  figure  in  units'  place  in  a  given  number  is  7.  What 
kind  of  numbers  will  not  divide  the  given  number? 

20.  Name  a  number  that  has  no  integral  factor  but  itself  and 
one.     What  kind  of  number  is  it? 

21.  Two  of  the  three  factors  of  a  number  being  given,  how 
can  the  remaining  factor  be  found  ? 


REVIEW  AND   PRACTICE  119 

22.  Reduce  to  simplest  form  : 

4  8     ^3_     12     8  A     4_0     14     15    3J>  Q. 
IV     9  '  ¥2'    10'     9  '   16'   25'  ^t' 

23.  Change  |  to  a  fraction  whose  denominator  is  81. 

24.  Change  S^^  ^^  ^^^  improper  fraction. 

25.  Divide  375  by  25. 

26.  What  is  the  cost  of  48  horses  at  $125  each? 

27.  The  average  price  per  dozen  paid  for  eggs  by  an  egg 
buyer  during  a  season  was  f  .16|.  At  that  rate,  what  did  he 
pay  for  1000  dozen?     How  many  eggs  could  he  buy  for  ilOO? 

28.  A  merchant  bought  700  yards  of  damaged  cloth  at  $.14| 
a  yard.  He  sold  200  yards  of  it  at  $.50  a  yard,  and  the  rest 
at  i.  10  a  yard.     How  much  did  he  gain? 

29.  Name  six  parts  that  a  bill  should  contain. 

30.  A  gallon  of  spirits  of  camphor  will  fill  how  many 
8-ounce  bottles? 

31.  A  stationer  bought  paper  at  f  1.00  a  ream  and  sold  it 
at  #.20  a  quire.     How  much  did  he  gain  on  10  reams? 

32.  How  many  degrees  are  there  in  all  the  angles  of  a 
rectangle  ? 

33.  Eighteen  straight  lines  are  drawn  from  the  center  to  the 
circumference  of  a  circle,  making  equal  angles  at  the  center. 
What  is  the  size  of  each  angle?  What  is  the  size  of  each  arc 
formed  in  the  circumference? 

34.  What  U.  S.  coin  is  most  nearly  like  the  English  shilling  ? 

35.  What  German  coin  is  most  nearly  like  the  U.  S.  25-cent 
piece? 

36.  What  is  the  silver  piece,  coined  in  this  country,  whose 
value  is  most  nearly  like  that  of  the  franc? 


120  GRAMMAR   SCHOOL   ARITHMETIC 

37.  What  is  the  area  of  a  triangle  whose  base  and  altitude 
are  respectively  25  rods  and  20  rods? 

38.  The  area  of  a  triangle  is  5  acres.  What  is  the  area  of 
a  parallelogram  having  the  same  base  and  altitude? 

39.  A  pile  of  stove-wood  is  12  ft.  long  and  8  ft.  high. 
What  is  it  worth  at  12.50  a  cord? 

40.  How  many  strips  of  carpet  27  in.  wide,  running  length- 
wise of  the  room,  are  required  to  carpet  a  room  9  ft.  wide  ? 

41.  A  piece  of  timber  1  ft.  square  and  20  ft.  long  contains 
how  man}^  board  feet? 

42.  The  volume  of  a  grain  bin  is  2,150,420  cubic  inches. 
How  many  bushels  of  grain  will  it  hold? 

43.  Make  and  solve  a  problem  that  requires  multiplication 
of  fractions. 

44.  Make  and  solve  a  problem  that  requires  reduction  of 
denominate  numbers. 

45.  Make  and  solve  a  problem  about  capacity  or  volume. 

46.  $15  worth  of  steel  wire  will  make  $1000  worth  of 
needles.  How  much  is  the  value  of  the  wire  increased  by  be- 
ing made  into  needles  ? 

47.  A  man  can  drill  60,000  needle-eyes  in  a  week.  That  is 
how  many  per  day?  How  many  per  hour,  if  he  works  eight 
hours  a  day? 

48.  If  750,000  medium-sized  needles  weigh  1  cwt.,  how  many 
would  it  take  to  make  a  pound  ? 

49.  112  sheets  of  14^'  x  20^'  IC  tin  roofing  plates  weigh  107 
pounds.  What  is  the  weight  of  560  such  plates?  Of  56  such 
plates  ? 


REVIEW   AND  PRACTICE 


121 


50.  (Ideas  of  Proportion.)  a.    10  is  how  many  times  2|- ? 

h.  If  21  quarts  of  berries  weigh  4|-  lb.,  what  will  10  quarts 
weigh  ? 

c.  How  many  quarts  will  weigh  9  lb  ? 

d.  What  will  15  quarts  weigh  ? 

51.  If  a  boy  can  carry  150  apples  weighing  3  ounces  apiece, 
how  many  apples  weighing  9  ounces  apiece  can  he  carry? 
IJ  ounces? 

251.     Written 

This  table,  compiled  from  the  records  of  the  United  States 
Weather  Bureau,  shows  in  inches  the  average  precipitation  of 
moisture  for  each  month  of  the  year  in  different  sections  of  the 
country. 


1 

g 

Si 

o 

.S5 

Of 

1 

4^ 

1 

1 
O 

1 

05 

a 

a 
'o 
!^ 

00 

e3 

1 

® 
> 

03 
CO 

1 

1 

< 

Jan. 

3.84 

1.72 

3.21 

3.28 

2.05 

4.53 

.65 

1.19 

.98 

.48 

1.33 

2.34 

4.33 

2.64 

Feb. 

3.50 

200 

2.99 

3.39 

1.64 

4.62 

.80 

1.08 

1.46 

.50 

1.40 

1.99 

4.03 

2.85 

Mar. 

4.27 

2.93 

2.70 

3.40 

1.27 

5.14 

1.77 

1.53 

2.09 

.91 

1.99 

1.44 

3.31 

2.87 

Apr. 

3.46 

2.11 

2.40 

2.89 

1.21 

4.98 

2.46 

3.00 

2.74 

1.98 

2.13 

1.29 

2.97 

1.15 

May 

3.45 

2.69 

3.14 

3.16 

2.77 

4.01 

3.34 

4.78 

5.28 

2.58 

1.97 

1.40 

2.26 

.49 

June 

3.02 

3.24 

3.52 

3.18 

4.14 

6.19 

3.75 

4.88 

4.76 

1.49 

.73 

1.48 

1.60 

.09 

July 

3.47 

3.11 

3.42 

4.19 

3.64 

6.36 

4.22 

3.83 

4.90 

1.65 

.52 

.69 

.80 

.01 

Aug. 

4.06 

3.35 

3.07 

4.50 

4.72 

5.68 

3.80 

3.57 

4.46 

1.36 

.74 

.50 

.50 

.03 

Sept. 

3.19 

2.82 

3.15 

3.41 

6.91 

4.63 

3.17 

2.99 

3.37 

.86 

.80 

.99 

2.12 

.08 

Oct. 

3.96 

2.98 

3.33 

3.01 

5.31 

2.9(i 

2.75 

2.75 

1.99 

.90 

1.50 

1.34 

2.96 

.81 

Nov. 

4.16 

2.04 

3.34 

3.18 

2.25 

3.74 

.98 

1.45 

1.08 

.53 

1.40 

2.27 

6.31 

1.35 

Dec. 

3.26 

2.27 

3.37 

2.95 

1.66 

4.25 

1.00 

1.36 

.93 

.64 

1.43 

2.40 

5.96 

2.99 

1-14.  Find,  to  the  nearest  hundredth,  the  average  monthly 
precipitation  in  each  of  the  cities  named.  Can  you  do  it  in 
30  minutes,  testing  your  work? 


122  GRAMMAR   SCHOOL   ARITHMETIC 

15.  The  population  of  the  Japanese  Empire  is  42,352,620,  and 
of  the  Russian  Empire  128,932,173  according  to  a  recent  cen- 
sus.    Find  the  difference  between  them. 

16.  The  earth,  in  its  revolution  around  the  sun,  passes 
through  space  at  the  rate  of  about  19  miles  a  second.  How- 
far  does  it  travel  during  a  30-minute  recitation  in  arithmetic? 

17.  a.  How  many  years  and  days,  taking  no  account  of  the 
extra  day  in  leap  year,  would  be  required  for  a  railroad  train, 
traveling  day  and  night  at  a  uniform  rate  of  50  miles  per  hour, 
to  travel  93,000,000  miles,  the  approximate  distance  from  the 
earth  to  the  sun? 

h.  The  planet  Neptune  is  about  thirty  times  as  far  from  the 
sun  as  the  earth  is.  Using  the  answer  to  «,  find  the  time  in 
which  such  a  train  could  travel  a  distance  equal  to  that  from 
the  sun  to  Neptune. 

18.  A  newspaper,  folded  into  four  leaves,  each  17^'  x  24"  in 
size,  has  seven  columns  on  a  page.  The  average  number  of 
copies  of  this  paper  printed  per  day  during  the  twenty-seven 
week-days  of  January,  1908,  was  48,400. 

a.    How  many  columns  were  printed? 

h.  If  all  these  papers  were  spread  out  in  single  sheets,  how 
many  acres  of  land  would  they  cover?     (Indicate  and  cancel.) 

19.  Multiply  2496  by  329  and  write  each  partial  product  in 
words. 


20.  (94.7  +  8.456  +  37.92  x  84  -  93.6  ^  1.8)  -- 14.4  -r-  .04. 

21.  The  roof  of  my  barn  is  sixty  feet  long.  The  slant  height, 
from  the  eaves  to  the  ridge,  is  25  feet  on  each  side.  It  is  cov- 
ered with  redwood  shingles  costing  $4.50  per  M.,  laid  4  inches 
to  the  weather.  7J  pounds  of  nails  were  used  with  each  thou- 
sand shingles  and  cost  §2.90  per  hundredweight.     The  men 


REVIEW   AND  PRACTICE  123 

who  laid  the  shingles  averaged  1350  shingles  per  day  for  each 
man,  and  received  13.00  each,  per  day. 

a.  What  did  the  shingles  cost  ? 
h.    What  did  the  nails  cost  ? 

c.  What  did  the  labor  cost  ? 

d.  What  did  the  roof  cost  ? 

22.  A  tile  roof  is  40  ft.  long  and  15  ft.  6  in.  from  eaves  to 
ridge  on  each  side. 

a.  What  was  its  cost  at  113.80  per  square  ? 
h.   What  is  the  weight  of  the  tile  in  tons,  if  975  lb.  of  tile 
will  make  a  square  of  roof  ? 

23.  A  factory  roof  is  made  of  sheets  of  tin  20''  by  28".  To 
make  the  seams,  2|  inches  are  taken  from  the  width,  and  |  of 
an  inch  from  the  length  of  each  sheet. 

a.  How  many  square  inches  of  roof  will  one  sheet  make  ? 
h.    How  many  sheets  will  make  a  square  of  roof  ? 

24.  Find  the  prime  factors  of  4503. 

25.  Determine  which  of  the  following  numbers  are  com- 
posite:    529,  403,  143,  397,  1943,  407. 

26.  Reduce  |^4f f  o  ^^  ^  fraction  whose  numerator  and  de- 
nominator are  prime  to  each  other. 

28;    Multiply  in  the  shortest  way: 

a.  8697  by  .331.  /.   4807  by  60,000. 

h.   9456  by  .25.  .  g.  817  by  25. 

c.  793,051  by  .142.  ^.  9796  by  .125. 

d.  6050  by  .125.  ^.    8796  by  16|. 

e.  39,764  by  99.  j.    74,583  by  111 


124                      GRAMMAR  SCHOOL 

ARITHMETIC 

29.    Divide  in  the  shortest 

way: 

a.  39,474  by  25. 

/.   42,835  by  14f. 

6.   9,726,250  by  125. 

ff,  7648  by  .25. 

e.   9438by33J. 

h.  93,042  by  .331 

d.  8753  by  .16§. 

^.    9843by.l4f 

e,   93,742  by  16f 

j.    86,728  by  16,000. 

30.  Add;  17f,  1911,  i|^  and  31|. 

31.  Find  the  smallest  number  that  will  exactly  contain  24, 
42,  54,  and  360. 

32.  Make  out  a  bill  containing  3  debit  and  2  credit  items, 
your  teacher  being  the  debtor  and  you  the  creditor.  Receipt 
the  bill  in  full  after  computing  the  balance. 

33.  The  surveyor  found  a  rectangular  piece  of  land  to  be 
2640  feet  long  and  880  feet  wide.  How  many  acres  did  it 
contain  ? 

34.  Find  the  number  of  seconds  in  a  solar  year. 

35.  How  many  fathoms  deep  is  the  ocean  at  a  place  where  a 
sounding  line  one  third  of  a  mile  long  will  just  reach  the  bottom  ? 

36.  Find  the  cost  of  a  flat  tin  roof  32'  x  24'  at  |9.60  per 
square. 

37.  a.  Find  the  exact  number  of  days  from  the  ninth  day  of 
last  January  to  the  present  time. 

b.  How  many  days  have  passed  since  the  last  Fourth  of  July  ? 
(?.  How  many  days  will  elapse  between  now  and  the  next 
Memorial  Day  ? 

38.  What  is  the  value  of  a  pile  of  uncut  building  stone, 
83'  by  6'  by  3',  at  16  per  cord  (99  cu.  ft.)? 

39.  It  requires  4  cu.  ft.  of  water  to  run  the  motor  of  our 
washing  machine  ten  minutes. 


COMPUTATION  IN  HUNDREDTHS  125 

a.    How  much  water  is  used  in  2  hr.  ? 

h.  If  the  motor  runs  two  hours  every  week,  what  is  the 
annual  cost  of  the  water  used,  at  14^  per  100  cu.  ft.  ? 

c.  How  many  gallons  of  water  are  used  ? 

d.  If  all  the  water  used  for  this  purpose  during  a  year 
were  collected  in  a  tank  12  ft.  by  13  ft.,  how  deep  would  the 
water  be  ? 

COMPUTATION  IN  HUNDREDTHS 

252.  Decimals  in  hundredths  are  used  very  generally  in 
business  calculations.  The  merchant  calculates  his  gain  or  loss 
as  a  certain  number  of  hundredths  of  the  cost  of  the  goods. 
Banks  compute  interest  in  hundredths.  Agents  who  sell  goods 
sometimes  figure  their  earnings  as  a  certain  number  of 
hundredths  of  the  selling  price  of  the  goods.  The  relations  of 
numbers  are  expressed  generally  in  hundredths. 

Problems  involving  computation  in  hundredths  usually 
present  one  of  the  two  questions  of  relation  between  product 
and  factors,  namely : 

a.    Two  factors  given,  to  find  the  product,  or, 

h.  The  product  and  one  factor  given,  to  find  the  other  fac- 
tor ;   e.g. : 

1.  A  merchant  bought  pears  at  11.60  a  bushel  and  sold  them 
so  as  to  gain  .25  of  the  cost.  How  much  did  he  gain  on  one 
bushel  ? 

Statement  of  Relation:  .25  of  $1.60  =  gain  on  one  bushel.  Here  1.60  and 
.25  are  factors,  and  the  product  is  to  be  found.     How  shall  we  find  it? 

2.  .40  of  the  pupils  in  a  school  are  boys.  If  there  are  600 
boys,  how  many  pupils  are  there  in  the  school? 

Statement  of  Relation :   .40  of pupils  =  600  pupils.     Here   600  is   a 

product  and  .40  one  of  its  factors.     How  may  the  other  factor  be  found? 


126  GRAMMAR   SCHOOL   ARITHMETIC 

3.  A  man's  salary  is  f  1500.  He  saves  1250.  How  many 
hundredths  of  his  salary  does  he  save  ? 

Statement  of  Relation:  of  $1500  =  ^250.    Here  250  is  a  product  and 

1500  one  of  its  factors.     How  may  the  other  be  found? 

253.     Written 

In  each  of  the  following  examples^  give  the  statement  of  relation 
and  find  the  answer : 

1.  A  farm  worth  14500  rents  for  .05  of  its  value.  For 
how  much  does  the  farm  rent? 

2.  .90  of  the  pupils  in  a  class  were  promoted.  If  36  pupils 
were  promoted,  how  many  were  there  in  the  class? 

3.  It  cost  124  to  decorate  a  room.  The  labor  cost  tl8. 
How  many  hundredths  of  the  entire  expense  were  for  labor? 

4.  A  farmer's  crop  of  apples  amounted  to  960  bushels,  qf 
which  864  bushels  were  fit  for  market.  How  many  hundredths 
of  the  crop  were  fit  for  market? 

5.  A  speculator  sold  some  property  for  178,000,  and  invested 
.33^  of  the  money  in  grain  and  f  39,000  in  real  estate.  He  put 
the  remainder  in  the  bank. 

a.    How  much  did  he  invest  in  grain? 

h.  How  many  hundredths  of  his  money  did  he  invest  in  real 
estate? 

c.    How  many  hundredths  of  his  money  were  left? 

6.  How  niany  dollars'  worth  of  goods  must  an  agent  sell  to 
earn  1513.40,  if  he  receives  .17  of  the  value  of  all  the  goods 
which  he  sells? 

7.  How  many  hundredths  of  1142.60  is  $7.13? 

8.  24  quarts  are  how  many  hundredths  of  six  bushels? 


COMPUTATION  IN  HUNDREDTHS         127 

9.  A  grocer  bought  8  bushels  of  potatoes  at  75  cents  a 
bushel  and  sold  them  for  17.80.  He  gained  how  many  hun- 
dredths of  the  cost? 

10.  Three  clerks  received  wages  as  follows  :  A,  115  a  week  ; 
B,  $10  a  week  and  .02  of  the  amount  of  his  sales;  C,  .05  of  the 
amount  of  his  sales.  What  was  each  clerk's  yearly  income,  if 
the  sales  of  each  amounted  to  $400  per  week? 

11.  .85  of  a  certain  number  is  595.  What  is  .14|^  of  the 
number  ? 

12.  A  boy  paid  .24  of  his  money  for  books,  .07  of  his  money 
for  stationery,  and  .22  of  his  money  for  a  football.  If  he  then 
had  13.76  left,  how  much  had  he  at  first? 

13.  Mr.  Markell  bought  a  house  for  $4200  and  sold  it  for 
$4830.    How  many  hundredths  of  the  cost  did  he  gain? 

14.  By  selling  his  automobile  for  $1860,  Dr.  Smith  received 
.66|  of  its  cost.    What  did  it  cost? 

15.  The  list  price  of  suits  for  a  baseball  team  was  $4.75 
apiece.  The  dealer  sold  11  suits  for  .80  of  the  list  price.  How 
much  did  he  receive  for  them? 

Note.  —  The  price  at  which  goods  are  marked  in  the  price  list  is  called 
the  list  price. 

16.  By  selling  goods  at  a  reduction  of  .15  of  the  list  price, 
a.    What  part  of  the  list  price  is  received? 

h.    What  is  the  list  price  of  goods  that  are  sold  for  $155.55? 
c.    What  reduction  is  made  on  goods  that  are  sold  for  $170? 

17.  A  contractor  makes  concrete  by  mixing  5  barrels  of 
cement,  10  barrels  of  sand,  and  25  barrels  of  crushed  stone. 
How  many  hundredths  of  the  mixture  is  :  a.  cement  ?  5.  sand  ? 
e,  crushed  stone  ? 


128  GRAMMAR   SCHOOL   ARITHMETIC 

PERCENTAGE 

254.  Per  cent  means  hundredths. 

Seven  per  cent  of  1100  means  .07  of  $100,  or  $7. 

Ten  per  cent  of  300  pounds  means  .10  of  300  lb.,  or  30  lb.     • 

Twenty-five  per  cent  of  24  hours  means  .25  of  24  hours,  or  6 
hours. 

Thirty-three  and  one  third  per  cent  of  276  means  .33^  of  276, 
or  92. 

The  sign  %  indicates  per  cent ;  e.g. 

19%  of  200  =  .19  of  200,  or  38. 
9%  of  ISO  =  .09  of  -f  80,  or  |7.20. 
^%  means  |^  of  1  %,  or  .001. 
I  %  means  |  of  1  %,  or  .OOf. 

255.  Oral 

Read  each  of  the  following  expressions,  using  the  word  hun- 
dredths instead  of  the  sign  ffo-i  and  find  its  value: 


1. 

9%  of  $300 

14. 

2f%  of  2100 

2. 

21  %  of  200 

15. 

125%  of  200 

3. 

75%  of  1000 

16. 

1%  of  200 

4. 

13%  of  30 

17. 

1  %  of  900 

5. 

44%  of  20 

18. 

f  %  of  1400 

6. 

89%  of  10001b. 

19. 

Jq  %  of  1000 

7. 

41%  of  lOObu. 

20. 

11%  of  1800 

8. 

96  %  of  10,000 

21. 

4|%  of  800 

9. 

3%  of  120 

22. 

15%  of  40 

10. 

621%  of  1000 

23. 

l3-\%  of  22 

11. 

130  %  of  100 

24. 

^%  of  70 

12. 

99%  of  2 

25. 

f%  of  140 

i3. 

6f%  of  800 

26. 

3J%  of  15 

PERCENTAGE 


129 


27.  4^\%  of  1100 

28.  66f%  of  11000 

29.  108%  of  11000 

30.  33  %  of  2  cents 

31.  2-1-  %  of  400 

32.  25  %  of  40 

33.  250  %  of  4 

34.  81%  of  30  days 


3|-%  of  15 


35.  6i%  of  12  feet 

36.  4|  %  of  1600  miles 

37. 

38.  100%  of  f  .37^ 

39.  -5^0%  of  1100 

40.  1%  of  18  quarts 

41.  I  %  of  40  sheep 

42.  j^  %  of  52  weeks 


256.  The  number  of  hundredths  indicated  as  per  cent  is  called 
the  rate  per  cent  ;  the  number  of  which  a  certain  number  of  hun- 
dredths are  indicated  by  the  rate  is  called  the  base  ;  the  product 
of  the  base  and  rate  is  called  the  percentage;  the  sum  of  the  base 
and  percentage  is  called  the  amount ;  the  difference  betweeyi  the 
base  and  percentage  is  called  the  difference  ;  e.g. 

25%  of  $300  is  175.  25%  is  the  rate  ;  1300  is  the  base; 
i75  is  the  percentage;  §375  is  the  amount;  ^225  is  the 
difference. 

257.  The  relations  of  product  and  factors  usually  determine 
the  method  to  be  employed  in  solving  problems  in  percentage  ; 
e.g. 

1.    A  man  bought  some  land  for  $4500,  and  sold  it  so  as  to 
gain  12  %  of  the  cost.     How  much  did  he  gain  ? 

Statement  of  Relation  :  12%  of  $4500  =  gain 
.12  of  $4500  3=? 


.12  and  $4500  are  factors,  and  the  product  is  to  be  found.     How  may  we 
find  it? 


130  GRAMMAR   SCHOOL   ARITHMETIC 

2.    A  house  rents  for  |630,  which  is  9%  of  its  value.     Find 
its  value. 


Statement  of  Relation :  9  %  of  (value  of  the  house)  = 

.09  of =    630 

630  is  a  product  aiid  .09  one  of  its  factors.  How  may  we  find  the  other 
factor? 

3.    A  merchant  gained  |19  on  an  article  that  cost  f  95.     The 
gain  was  what  per  cent  of  the  cost  ? 
Statement  of  Relation :  — -  of  $95  =  ^19 
19  is  a  product  and  95  one  of  its  factors.     Find  the  other  factor. 

258.  In  finding  the  rate  per  cent,  or  number  of  hundredths, 
how  many  decimal  places  must  there  be  in  the  quotient  ?  Then 
how  must  the  number  of  decimal  places  in  the  dividend  com- 
pare with  the  number  of  decimal  places  in  the  divisor  ? 

Summary 

Before  dividing,  to  find  the  rate  per  cent,  arrange  the  dividend 
and  divisor  so  that  the  dividend  contains  two  more  decimal  places 
than  the  divisor.  This  may  he  done  hy  annexing  ciphers  to  one  or 
the  other  of  these  terms,  as  may  he  necessary. 

If  the  quotient  is  not  exact  when  two  decimal  places  have  heen 
reached,  express  the  remainder  as  a  common  fraction,  i7i  the  quo- 
tient, thus  : 

a.    7  bushels  are  what  per  cent  of  8.5  bushels  ? 

Statement  of  Relation: of  8.5  bu.  =  7  bu. 

^2f  §  =  .  82^^  or  82fy  % .     Ans, 

8.5)7.0-00 
680 
200 
170 
30 


PERCENTAGE 


131 


5.  A  lake  in  Maine  is  152.875  rods  long  and  92  rods  wide. 
Its  length  is  what  per  cent  of  its  width  ? 

Statement  of  Relation : •  of  92  rd.  =  152.875  rd. 

1.66l|f  =  I.QQjW  or  166-jV^ % .     Ans. 
92.0  j  152.875 
920 
6087 
5520 
5675 
5520 
155 

Note.  —  Care  should  be  taken  to  express  the  decimal  rate  per  cent  prop- 
erly, as  hundredths.  Every  fractional  part  of  l^o  must  be  written  at  the 
right  of  the  hundredths'  place. 


1%=    .01 

121%  =.121 

or 

.125 

9%=    .09 

i%  =  .00l 

or 

.005 

10%=    .10 

io^V%  =  -ioA 

or 

.107 

90%=    .90 

331%  =  .331 

100%  =  1.00 

8i%  =  .08i 

or 

.0825 

900%  =  9.00 

i%  =  .ooi 

or 

.0025 

125%  =1.25 

J%  =  .00i 

or 

.00125 

21 

)9.    Written 

1. 

Express  decimally : 

«•  7%               /.  61% 

k,  101% 

P'  i% 

5.   6%                g,   121% 

I.  110% 

q^  i% 

c.  2%                h.  15|% 

m.  2b^(fo 

r^   i% 

d.  12%              ^.  37|% 

n.  200% 

s,  f  % 

e.  78%              y.  4f% 

0.  1271% 

t  iV/ 

2. 

1291  is  16|%  of  what? 

3. 

35  %  of  a  number  is  700. 

Find  the  number. 

132  GRAMMAR   SCHOOL   ARITHMETIC 

4.  84.20  is  what  per  cent  of  421? 

5.  Find  I  %  of  $5600. 

6.  Find66f%  of  927  tons. 

7.  39.744  is  what  per  cent  of  900  ? 

8.  A  short  ton  is  what  per  cent  of  a  long  ton  ? 

9.  386%  of  244=  what? 

10.  23^  %  of  a  number  is  7000.     Find  the  number. 

11.  A  nautical  mile  is  6086.07  feet.  A  statute  mile  is  what 
per  cent  of  a  nautical  mile  ? 

12.  Find  a  number,  |  per  cent  of  which  is  287. 

13.  48  rods  are  what  per  cent  of  a  mile  ?  * 

14.  57 1  cubic  inches  are  what  per  cent  of  one  gallon  ? 

15.  Find  S^\%  of  $235. 

16.  17  %  of  2475  is  what  per  cent  of  720  ? 

17.  .043  is  what  per  cent  of  17.2? 

18.  A  man's  salary  is  $1850  and  his  expenses  $1757.50. 
His  expenses  are  what  per  cent  of  his  salary  ? 

19.  A  man's  expenses  are  $2140  a  year.  His  salary  is 
125%  of  this  sum.     Find  his  salary. 

20.  A  man  bequeathed  18  %  of  his  estate  to  a  hospital,  7  % 
to  a  missionary  society,  and  30  %  to  his  wife.  The  remainder 
was  divided  equally  among  his  three  brothers.  If  the  estate 
amounted  to  $72,600,  how  much  did  each  of  the  brothers 
receive  ? 

21.  For  how  much  a  month  must  a  house  w^orth  $6000  be 
rented  in  order  that  the  rent  may  amount  to  7|  %  of  the  value 
of  the  house  ? 


PER  CENTS  EQUIVALENT   TO   COMMON  FRACTIONS        133 

22.  A  grocer  bought  12  cases  of  coffee,  each  containing  50 
one-pound  packages,  and  sold  240  packages.  What  per  cent  of 
the  coffee  did  he  sell  ? 

23.  Which  is  greater,  and  how  much,  50%  of  75,  or  75% 
of  50? 

24.  A  field  is  375  feet  long  and  150  feet  wide. 
a.   Its  breadth  is  what  per  cent  of  its  length? 
h.   Its  length  is  what  per  cent  of  its  breadth? 

25.  Find  in  acres  the  area  of  a  rectangular  field  whose  width 
is  45  rods  and  whose  length  is  142|  %  of  its  width. 

26.  A  manufacturing  company  employing  272  persons,  whose 
weekly  wages  average  f  12  apiece,  raises  the  wages  of  its  em- 
ployees 8^  % .     How  much  per  year  is  then  paid  to  all  of  them  ? 

27.  18  %  of  the  men  in  an  army  died  of  disease.  If  the  loss 
from  this  cause  was  1260  men,  how  many  men  were  there  in 
the  army  at  first? 

28.  A  man  having  1 20,000  in  the  bank  drew  out  30  %  of  it 
and  then  25  %  of  what  was  left.  How  many  dollars  still  re- 
mained in  the  bank? 

29.  83  %  of  the  boys  in  a  military  school  attended  a  game 
of  football.  If  166  boys  attended  the  game,  how  many  boys 
were  there  in  the" school? 

PER   CENTS   EQUIVALENT   TO   COMMON   FRACTIONS 

260.  All  percentage  problems  involving  the  relation  of  prod- 
uct and  factors  may  be  solved  in  decimals.  But  in  many  cases 
the  work  may  be  shortened  by  changing  the  per  cents  to  com- 
mon fractions. 

261.  Oral 

1.   The  whole  of  anything  is  how  many  hundredths  of  it  ? 
What  per  cent  of  it? 


134  GRAMMAR   SCHOOL   ARITHMETIC 

2.  ^  of  anything  is  how  many  hundredths  of  it?     J?     ^? 

1  ?   3?   2?   3.?   4?   1?   3?   5  9  X?  1?  2?  1?  5?  _1_?   JL? 
T¥  •   4  •  'E-      "5  •   5  •   ^  •   ¥  •   t  •   8  •   3  •   3  '   6  '   6  '   12-   20* 

3.  What  common  fraction  is  the  same  as  .10?  .20?  .30? 
.40?  .50?  .60?  .70?  .80?  .90?  .25?  .33|?  .14f?  621? 
.371?     .66|?     .121?    .871?    .75?    .16|?     .831?    .081.     .05? 

4.  What  per  cent  is  the  same  as  ^?     ^?    -J?    -J?    ^?    |?    ^-? 

JL?   _1_?  JL?  JL?  JL?   2?   3?   2.?   3?   4?   3?   4?   7? 
10-   12-  20-  16-  25-   3-   ?•   5*   5'   5'   ¥'   S'   t' 


5.  Learn  thii  table : 

J=50% 

i=8H% 

1  =  621% 

i  =  25% 

|  =  66|% 

1  =  871% 

1  =  76%  , 

i  =  16f% 

tV  =  io% 

-H20% 

1  =  831% 

^=^% 

1  =  40% 

j  =  14f% 

2^  =  5% 

f=60% 

i  =  121% 

iV  =  6i% 

i  =  80% 

1=371% 

A  =  4%' 

6.  What  per  cent  of  anything  is  left  after  50%  of  it  has 
been  taken  away?  After  75%  of  it  has  been  taken  away? 
30%?  40%?  35%?  331%?  621  %?  371  %?  831  %?  16|%? 
49%?     1%?     21-%?     981%? 

7.  What  per  cent  of  anything  is  left  after  15%,  10%,  and 
5  %  of  it  have  been  taken  away? 

8.  A  girl  used  12  %  of  her  Christmas  money  on  one  day,  18  % 
the  next  day,  and  15  %  the  next  day.  What  per  cent  of  her 
money  remained? 

9.  97|-  per  cent  of  the  pupils  belonging  to  a  certain  school 
were  present.     What  per  cent  of  the  pupils  were  absent? 

10.    2-|  %  of  the  pupils  in  a  school  were  absent.     What  per 
cent  of  the  pupils  were  present? 


PERCENTAGE  135 

11.  37  5^  of  a  shipment  of  peaches  were  spoiled.     What  per 
cent  of  the  peaches  were  good? 

12.  Using  common  fractions  instead  of  decimals^  find  : 


a. 

50%  of  $124 

i- 

371%  of  64  days 

I, 

25%  of  36 

h. 

871%  of  72  pounds 

c. 

75%  of  24 

I 

81  %  of  132  square  miles 

d. 

331%  of  999 

m. 

5%  of  1200,000 

e. 

66|  %  of  42 

n. 

6J%  of  32  quarts 

/. 

831%  of  30 

0. 

4  %  of  75  cents 

9- 

16|%  of  48  bushels 

P- 

371%  of  56  minutes 

h. 

14|-%  of  49  feet 

<1' 

16|  %  of  180  grains 

i. 

121%  of  30  inches 

r. 

121%  of  64  miles 

13.  5  cents  are  12|^%  of  what  sum? 

14.  Frank  paid  16|%  of  his  money  for  a  book  that  cost  f  .20. 
How  much  money  had  he  ? 

15.  Arthur  earned  $1.60  and  paid  75%  of  it  for  a  football. 
How  much  had  he  left  ? 

16.  Wallace  earned  a  sum  of  money,  paid  87 J  %  of  it  for  a 
suit  of  clothes,  and  had  |2  left. 

a.    $2  is  what  per  cent  of  the  money  which  he  earned? 
h.    How  much  did  he  earn  ? 

17.  A  farmer  raised  a  crop  of  potatoes,  sold  QQ^  %  of  them, 
and  had  200  bushels  left. 

a.    200  bushels  were  what  per  cent  of  the  crop  ? 
h.    How  many  bushels  were  raised  ? 
c.    How  many  bushels  were  sold  ? 

18.  One  morning  Lucy  cut  a  basket  of  roses ;   25  %  of  them 
were  pink,  331  %  yellow,  16|  %  red,  and  the  rest  white. 

a.    What  per  cent  of  the  roses  were  not  white .? 


136  GRAMMAR   SCHOOL   ARITHMETIC 

h.  What  per  cent  were  white  ? 

c.  If  there  were  15  white  roses,  how  many  roses  did  Lucy 
cut? 

d.  How  many  were  yellow  ? 

e.  How  many  were  pink  ? 
/.    How  many  were  red  ? 

19.  A  wholesale  grocer  hought  a  carload  of  flour.  He  sold 
121  %  of  it  to  A,  30  %  to  B,  37^  %  to  C,  and  the  remainder, 
which  was  48  barrels,  to  D. 

a.  What  per  cent  of  the  flour  did  he  sell  to  D  ? 

b.  How  many  barrels  of  flour  did  the  carload  contain  ? 

20.  A  and  B  hired  a  horse.  B  paid  66^%  of  the  expense 
and  A  the  remainder,  which  was  f  1.50.  What  was  the  entire 
expense  ? 

21.  Three  coal  dealers  furnished  the  coal  for  the  schools  of  a 
city.  The  first  furnished  50  %  of  it,  the  second  1400  tons,  and 
the  third  16f  %  of  it. 

a.  What  per  cent  of  the  coal  did  the  second  dealer  furnish  ? 

b.  How  many  tons  did  all  furnish  ? 

c.  How  many  tons  did  the  first  furnish  ? 

22.  A  man  bought  a  lot  and  built  a  house  on  it.  The  lot 
cost  $800,  which  was  50  %  of  the  cost  of  the  house.  How  much 
did  both  cost  ? 

23.  I  of  a  farm  is  cultivated.  What  per  cent  of  it  is 
uncultiv.ated  ? 

24.  A  pole  stands  in  a  pond  of  water  so  that  -|  of  the  length 
of  the  pole  is  in  the  mud,  and  |  in  the  water.  What  per  cent 
of  the  length  of  the  pole  is  above  the  water  ? 

25.  25  %  of  60  added  to  |  of  60  equals  what  ? 

26.  Edward  solved  16  problems  and  93|%  of  them  were 
correct.     How  many  were  incorrect  ? 


PERCENTAGE  137 

27.  50  %  of  80  %  of  anything  is  what  per  cent  of  it  ? 

28.  80  %  of  50  %  of  anything  is  what  per  cent  of  it  ? 

29.  100  %  of  anything  added  to  25  %  of  it  equals  what  per 
cent  of  it  ? 

30.  A  man  had  a  sum  of  money  and  gained  a  sum  equal  to 
40%  of  what  he  had  at  first.  He  then  had  what  per  cent  of 
the  first  sum  ? 

31.  20%  more  than  $5  is  what  per  cent  of  f  5  ? 

32.  20%  more  than  f  5  is  how  many  dollars? 

33.  50  %  more  than  f  8  is  how  many  dollars  ? 

34.  1250  is  125%  of  what  sum  ? 

35.  f  250  is  25  %  more  than  what  sum  ? 

36.  A  colt  worth  $100  increased  25%  in  value  in  eight 
months.     How  much  was  it  then  worth  ? 

37.  A  liveryman  bought  a  horse  for  $200.  Its  value  de- 
creased 50  %  in  one  year.     What  was  it  then  worth  ? 

38.  What  number,  increased  by  50  %  of  itself,  equals  300  ? 
262.    Written 

1.  Find  : 

a.  12i%  of  896  bu.  e.    |%  of  $15,000 

h.  66f  %  of  927  T.  /.  5%  of  15,000 

c.  87-1%  of  240  gal.  g.    500%  of  15,000 

d.  16f  %  of  636  qt.  h.    130%  of  480 

2.  39.40  is  16f%  of  what? 

3.  3.27  is  what  per  cent  of  8.72? 

4.  8.72  is  what  per  cent  of  3.27  ? 

5.  3121  is  62^  %  of  a  certain  number.    What  is  Sl^  %  of  the 
same  number  ? 


138  GRAMMAR  SCHOOL  ARITHMETIC 

6.  Express  decimally  J,  i%,  |,  |.%,  |,  i%,  f,  |%,  f,  \% 

7.  A  floor  is  16  ft.  square.  What  per  cent  of  it  may  be 
covered  by  a  rug  that  is  3  yd.  long  and  2  yd.  wide  ? 

8.  A  man  paid  %  175  for  a  horse,  %  125  for  a  wagon,  and  f  25 
for  a  harness.  What  per  cent  of  the  entire  cost  was  the  cost  of 
each  ? 

9.  From  a  field  containing  40  A.,  a  rectangular  piece  40  rd. 
long  and  20  rd.  wide  was  sold.  What  per  cent  of  the  field 
remained  ? 

10.  A  man  withdrew  35%  of  his  deposits  from  the  bank, 
leaving  $3250  in  the  bank.  How  many  dollars  did  he 
withdraw  ? 

11.  Robert  attended  school  86  days  during  a  term  and  was 
marked  95f  %  in  attendance.  How  many  days  were  there  in 
the  term  ? 

12.  A  piece  of  cloth  shrank  4  %  in  sponging,  after  which  it 
contained  48  yd.  How  many  yards  did  the  piece  contain 
before  it  was  sponged  ? 

13.  A  telephone  was  placed  in  my  house  on  the  20th  day  of 
October,  1907.  Beginning  with  that  day,  I  had  the  use  of  the 
telephone  what  per  cent  of  the  year  1907  ? 

14.  A  telephone  company  increased  its  charge  from  $30  a 
year  to  $  36  a  year.     What  was  the  rate  per  cent  of  increase  ? 

15.  A  depositor  withdrew  40  %  of  his  balance  at  the  bank, 
and  bought  a  piece  of  furniture  for  %  72,  which  was  12  %  of  the 
sum  withdrawn. 

a.  What  was  his  balance  before  withdrawing  ? 

h.    What  per  cent  of  it  did  he  pay  for  furniture  ? 

c.    How  many  dollars  of  his  withdrawal  remained  unused  ? 


PROFIT   AND   LOSS  139 

16.  My  city  tax  bill  amounts  to  $82.60  this  year.  If  I  do 
not  pay  it  before  Nov.  15,  two  per  cent  will  be  added  to  the 
amount  of  the  bill.     What  must  I  pay  if  I  wait  till  Nov.  16  ? 

17.  I  obtained  a  5%  reduction  by  paying  my  semi-annual 
water  bill  before  the  15th  of  July.  The  bill,  after  being 
reduced,  was  12.375.  What  was  it  at  first?  How  much  did 
I  have  to  pay? 

18.  The  population  of  a  city  has  increased  12  %  in  the  last 
three  years.     It  is  now  112,000.     What  was  it  three  years  ago  ? 

19.  The  €ost  of  living  was  estimated  to  be  40  %  greater  in 
1907  than  in  1897. 

a.  Assuming  this  estimate  to  be  correct,  how  much  money 
was  necessary  to  support  in  1907  such  a  family  as  was  sup- 
ported for  11250  in  1897? 

h.  A  workingman's  wages  of  %  15  a  week  in  1897  were  equiva- 
lent to  what  sum  per  week  in  1907  ? 

c.    A  salary  of  %  1820  in  1907  was  equal  to  what  in  1897  ? 

20.  The  daily  sales  of  a  department  store  last  year  averaged 
I  8262,  which  was  2  %  greater  than  the  daily  average  for  the 
year  before.     What  was  the  daily  average  for  the  year  before  ? 

PROFIT  AND  LOSS 

263.  When  property  is  sold  for  more  or  less  than  it  cost,  the 
gain  or  loss  is  always  computed  as  a  certain  per  cent  of  the  cost. 

Each  of  the  following  expressions,  when  used  in  a  problem,  means  that 
the  profit  or  gain  is  10  %  of  the  cost : 

At  a  profit  of  10  % ;  at  10  %  gain ;  at  10  %  above  cost ;  at  an  advance  of  10  %. 

264.  Oral 

1.    A  book  that  cost  $5  was  sold  at  a  gain  of  25%.     What 
was  the  gain? 

Statement  of  Relation:  25%  of  |5  =  gain.  •. 

Which  term  of  relation  (factor  or  product)  is  to  be  found? 


140  GRAMMAR  SCHOOL   ARITHMETIC    - 

2.  A  grocer  paid  80  cents  a  bushel  for  potatoes  and  sold 
them  at  a  profit  of  20  cents  a  bushel.  What  per  cent  did  he 
gain? 

Statement  of  Relation :  %  of  |  .80  =  $  .20. 

Which  term  of  relation  is  to  be  found  ? 

3.  A  furniture  dealer  sold  a  desk  at  a  gain  of  25%.  He 
gained  $5.     What  did  the  desk  cost? 

Statement  of  Relation :  25%  of  cost  =  |5. 
Which  term  of  relation  is  to  be  found  ? 

4.  The  whole  of  anything  is  what  per  cent  of  it  ? 

If  the  cost  of  an  article  is  100  %  of  the  cost,  and  the  gain  is 
10%  of  the  cost,  the  selling  price,  which  is  the  sum  of  the  cost 
and  the  gain,  is  what  per  cent  of  the  cost? 

5.  An  article  that  cost  $8  was  sold  at  a  gain  of  10%.  Find 
the  selling  price. 

Statement  of  Relation:  110%  of  $8  =  selling  price. 
Which  term  of  relation  is  to  be  found? 

6.  A  fruit  dealer  lost  40%  on  a  shipment  of  peaches  that 
cost  him  $200.     How  much  did  he  lose?    What  did  he  receive? 

7.  A  produce  dealer  sold  potatoes  at  $2.20  a  barrel,  thereby 
gaining  10%.     What  did  they  cost  per  barrel? 

Statement  of  Relation :  110%  of  cost  =  |2.20. 
Which  term  of  relation  is  to  be  found  ? 

8.  A  man  sold  his  farm  at  $32  per  acre,  thereby  losing  20%. 
What  price  per  acre  did  he  pay  for  the  farm  ? 

100%)  of  the  cost  less  20%,  of  the  cost  =  what  per  cent  of  the  cost? 
Statement  of  Relation :  80%  of  the  cost  =  $32. 

■  Which  term  of  relation  is  to  be  found  ? 


PROFIT  AND  LOSS  141 

9.  An  article  that  cost  $200  was  sold  at  a  gain  of  50  %. 

a.  What  was  the  selling  price? 

h.  What  was  the  gain? 

10.  On  an  article  that  sold  for  $  180  the  dealer  lost  10  %. 
a.  What  was  the  cost? 

h.    How  much  was  lost? 

11.  On  an  article  that  sold  for  $2.40  the  dealer  gained  20%. 
a.    What  was  the  cost? 

h.    What  was  the  gain? 

12.  Cloth  that  cost  $2  a  yard  was  sold  for  |3  a  yard. 
a.    What  was  gained  on  a  yard  ? 

h.    What  per  cent  was  gained? 

13.  A  dealer  bought  hops  at  40^  a  pound  and  sold  them  at 
30/  a  pound. 

a.    What  was  the  loss  on  a  pound? 
h.    What  per  cent  was  lost? 

14.  What  per  cent  was  gained  on  a  city  lot  bought  for  1400 
and  sold  for  1500? 

15.  What  per  cent  was  lost  on  a  city  lot  bought  for  $  500 
and  sold  for  $400? 

16.  At  what  price  must  goods  costing  $7.20  be  sold  to  yield 
a  profit  of  16|  %  ? 

17.  $1  profit  on  a  pair  of  shoes  costing  $4  is  what  per  cent 
profit? 

18.  A  profit  of  $1  on  a  pair  of  shoes  sold  for  $4.00  is  what 
per  cent  profit  ? 

265.     Written 

1.    a.  What  is  the  profit  on  1  ton  of  pork  bought  at  $7.50 
per  hundredweight  and  sold  at  $.10  per  pound? 
h.    What  is  the  rate  of  profit  ? 


142  GRAMMAR  SCHOOL   ARITHMETIC 

2.  A  contractor  gained  12-1  %  on  a  job  of  grading  that  cost 
him  $2448.     How  many  dollars  did  he  gain  ? 

3.  A  carriage  dealer  gained  18%  by  selling  a  carriage  for 
$36  more  than  he  paid  for  it.     Find  its  cost. 

4.  a.  What  must  a  grocer  receive  per  barrel  for  flour,  in 
order  that  he  may  make  a  profit  of  22|  %  on  flour  that  costs 
$4.50  per  barrel? 

h.    What  is  his  gain  on  75  barrels? 

5.  A  stock  of  paper  costing  $2345  was  damaged  by  water  so 
that  it  had  to  be  sold  at  a  loss  of  15  %.  What  was  the  selling 
price  ? 

6.  a,  A  grocer  selling  sugar  at  $5.50  per  hundredweight 
makes  a  profit  of  10%.  How  much  per  ton  does  the  sugar 
cost  him  ? 

h.    How  much  does  he  gain  on  7  T.  of  sugar? 

c.    How  many  pounds  must  he  sell  in  order  to  gain  $25  ? 

7.  A  hardware  merchant  bought  75  hundred-pound  kegs  of 
nails  for  $206.25. 

a.  When  he  sells  them  at  3J^  a  pound,  what  per  cent  profit 
does  he  make  ? 

h.  When  he  sells  them  at  $2.90  per  keg,  what  per  cent  profit 
does  he  make  ? 

c.  At  what  price  per  keg  must  he  sell  them  to  make  a  profit 
of  16%? 

8.  The  proprietor  of  a  market  received  a  shipment  of  600  lb. 
of  hams,  costing  $15  per  hundredweight.  He  allowed  for  a 
shrinkage  of  10  lb.  while  they  were  being  sold,  and  marked 
them  so  as  to  gain  31^%.  At  what  price  per  pound  did  he 
mark  them  ? 


PROFIT   AND   LOSS  143 

9.   Mr.  Jennings  sold  his  automobile  for  $2142,  thereby  los 
ing  16%.     What  did  it  cost?     Make  and  solve  another  prob- 
lem based  on  the  facts  given  in  this  problem. 

10.  What  per  cent  is  gained  on  carpets  bought  at  90  cents  a 
yard  and  sold  at  ^1.25  a  yard  ?  Make  and  solve  another  prob- 
lem based  on  the  facts  given  in  this  problem. 

11.  A  grocer  makes  a  profit  of  10  %  by  selling  sugar  at  50 
cents  per  hundredweight  above  cost.  At  what  price  per  pound 
does  he  sell  it  ?  Make  and  solve  another  problem  based  on  the 
facts  given  in  this  problem. 

12.  Hats  that  cost  1 27  a  dozen  were  sold  for  13.50  apiece. 
What  was  the  rate  per  cent  of  profit  ?  Make  and  solve  another 
problem  based  on  the  facts  given  in  this  problem. 

13.  A  man  bought  a  city  lot  for  1 2400  and  sold  it  so  as  to 
gain  20%.     How  much  did  he  receive  for  the  lot? 

14.  A  man  sold  a  house  and  lot  for  12400,  thereby  gaining 
20  % .     How  much  did  the  lot  cost  ? 

15.  By  selling  ahorse  for  |189  the  owner  lost  10%.  At 
what  price  must  he  have  sold  the  horse  to  gain  10  %  ? 

16.  A  horse  dealer  bought  a  span  of  horses  for  1240  apiece. 
He  sold  them  so  as  to  gain  20  %  on  one  and  lose  20  %  on  the 
other.     What  was  his  gain  or  loss  by  the  transaction  ? 

17.  A  jeweler  sold  two  watches  for  $60  apiece.  He  gained 
20  %  on  one  and  lost  20  %  on  the  other. 

a.    How  much  did  he  gain  or  lose  by  the  transaction? 
h.    What  per  cent  did  he  gain  or  lose  by  the  transaction  ? 

18.  A  merchant  sells  goods  at  an  average  profit  of  30%. 
60  %  of  his  goods  are  sold  for  cash  and  the  remainder 
are  sold  on  credit.  He  loses  5%  of  his  credit  sales  in  bad 
debts. 


144  GRAMMAR  SCHOOL   ARITHMETIC 

a.  How  much  cash  does  he  receive  for  a  stock  of  goods  that 
cost  $36,000? 

h.  How  many  dollars  does  he  charge  on  his  books  from  the 
sale  of  this  stock  ? 

c.  How  much  does  he  lose  in  bad  debts  ? 

d.  What  is  his  net  gain  ? 

e.  What  per  cent  does  he  gain,  making  allowance  for  bad 
debts  ? 

19.  A  huckster  buys  sweet  corn  at  il.25  per  hundred  ears 
and  sells  it  at  20^  a  dozen. 

a.    What  per  cent  profit  does  he  make  ? 

h.  At  what  price  per  dozen  must  he  sell  it  in  order  to  make 
a  profit  of  40  %  ? 

c.  How  many  ears  must  he  sell  at  an  advance  of  10  %  in 
order  to  gain  13.00? 

20.  A  farmer  bought  a  piece  of  land,  and,  after  keeping  it 
a  number  of  years,  desired  to  sell  it.  He  asked  40%  more 
than  he  paid  for  it,  and  then  sold  it  for  $3780,  which  was  90% 
of  his  asking  price. 

a.   What  was  his  asking  price  ? 

h.   What  did  the  farmer  pay  for  the  land  ? 

c.  How  much  did  he  gain  ? 

d.  What  per  cent  did  he  gain  ? 

e.  For  how  much  should  he  have  sold  the  land  to  gain  16|  %  ? 

21.  A  manufacturer  sold  his  goods  at  60  %  above  the  actual 
cost  of  manufacture,  and  was  able  to  collect  only  96  %  of  his 
sales.     He  collected  f  18,000  from  one  month's  sales. 

Make  and  solve  four  problems,  using  these  facts. 

22.  Make  and  solve  a  problem  that  requires  the  gain  to  be 
found. 


COMMISSION  145 

23.  Make  and  solve  a  problem  that  requires  the  rate  per  cent 
of  loss  to  be  found. 

24.  Make  and  solve  a  problem  that  requires  the  cost  to  be 
found. 

25.  Make  and  solve  a  problem  that  requires  the  rate  per  cent 
of  gain  to  be  found. 

26.  Make  and  solve  a  problem  that  requires  the  selling  price 
to  be  found. 

COMMISSION 

266.  One  who  transacts  business  for  another  is  an  agent. 
Agents  are  known  by  various  names  according  to  the  kind 

of  business  transacted  by  them.  Those  who  buy  and  sell  mer- 
chandise on  commission  are  called  commission  merchants  or 
commission  brokers ;  those  who  buy  and  sell  stocks  and  bonds 
are  called  stock  brokers;  those  who  collect  money  are  called 
collectors.     Can  you  mention  other  kinds  of  agents  ? 

267.  The  percentage  allowed  an  agent  as  compensation  for 
transacting  business  is  called  commission. 

268.  The  commission  of  a  broker  is  called  brokerage. 

269.  Commission  for  buying  goods  is  computed  as  a  certain 
per  cent  of  the  cost  of  the  goods;  commission  for  selling  goods  is 
computed  as  a  certain  per  cent  of  the  selling  price  of  the  goods; 
commission  generally  is  computed  as  a  certain  per  cent  of  the 
money  handled,  or  the  value  of  the  property  with  which 
the  agent  deals.  The  principal  exception  to  this  rule  is 
brokerage  for  buying  and  selling  stocks  and  bonds,  which  will 
be  treated  later. 

270.  A  quantity  of  goods  delivered  to  a  commission  merchant 
to  be  sold  is  called  a  consignment. 


146  GRAMMAR  SCHOOL   ARITHMETIC 

271 .  The  party  sending  a  consignment  of  goods  to  he  sold  hy  a 
commission  merchant  is  the  consignor. 

272.  The  party  to  whom  a  consignment  of  goods  is  delivered 
for  sale  is  the  consignee. 

273.  The  sum  received  from  the  sale  of  goods^  after  all  expenses^ 
such  as  commission^  freight^  and  cartage^  have  been  deducted^  is 
called  the  net  proceeds  of  the  sale. 

274.  The  party  who  employs  an  agent  is  called  the  principal. 

275.  Oral 

1.  A  college  student  sold  200  books  at  $3  apiece  during  a 
summer  vacation.     What  was  his  commission,  at  40  %? 

2.  A  real  estate  agent  received  180  for  selling  a  house.  His 
commission  was  2  % .     What  was  the  selling  price  of  the  house  ? 

Statement  of  Relation :  2  %  of =  $  80. 

Which  term  of  relation  is  to  be  found  ? 

3.  A  lawyer  received  |30  for  collecting  $200.  What  was 
the  rate  of  his  commission? 

Statement  of  Relation:  %  of  $200  =  $30. 

Which  term  of  relation  is  to  be  found  ? 

4.  A  commission  merchant  sold  1000  pounds  of  butter  at  25 
cents  a  pound,  retained  his  commission  of  10%,  and  sent  the 
remainder  to  his  principal. 

a.  What  did  his  commission  amount  to? 

b.  How  much  did  the  principal  receive? 

5.  An  auctioneer  sold,  on  10  %  commission,  household  goods 
to  the  amount  of  $100.  What  were  the  net  proceeds  of  the 
sale? 

6.  When  an  agent  sells  goods  on  20  %  commission,  what  per 
cent  of  the  selling  price  of  the  goods  does  the  principal  receive  ? 


COMMISSION  147 

7.  A  manufacturing  company  sold  its  entire  product  through 
a  commission  merchant  who  received  10%.  What  was  the 
selling  price  of  a  consignment  for  which  the  company  received 

1900? 

8.  The  net  proceeds  of  a  sale  were  $85.  The  commission 
was  flo.     What  was  the  rate  of  commission? 

9.  What  rate  of  commission  is  received  when  a  sale 
amounting  to  $100  yields  1 80  net  proceeds? 

10.  A  commission  merchant  receives  2  cents  a  dozen  as  his 
compensation  for  selling  eggs. 

a.  That  is  equivalent  to  what  per  cent  commission  when 
eggs  sell  at  20  cents  a  dozen? 

h.    When  they  sell  at  16  cents  a  dozen? 
c.    When  they  sell  at  24  cents  a  dozen? 

11.  A  collector  for  a  daily  newspaper  received  5  %  commis- 
sion. How  much  must  he  collect  daily  in  order  to  earn  $4  a 
day? 

12.  A  collector  working  on  10%  commission  must  collect 
how  many  dollars  in  order  that  his  principal  may  receive  1180? 

13.  An  agent  collected  a  sum  of  money,  took  out  his  com- 
mission of  20  %,  and  paid  the  remainder,  which  was  $40,  to  his 
employer.     What  was  his  commission  ? 

276.     Written 

1.  What  is  an  agent's  commission  at  4J  %  for  selling  850 
barrels  of  flour  at  $5.25  a  barrel  ? 

2.  A  commission  merchant  sold  a  consignment  of  goods  for 
$2470,  took  out  his  commission  of  8%,  paid  $28  freight  and  $5 
storage,  and  sent  the  remainder  to  the  consignor.  How  much 
did  the  consignor  receive  ? 


148  GRAMMAR  SCHOOL   ARITHMETIC 

3.  An  agent  receives  6  %  commission  for  buying  wool  at 
21  cents  a  pound. 

a.  What  is  his  commission  for  buying  50  tons  of  wool  ? 
h.    How  many  pounds  must  he  buy  in  order  to  earn  $1690.50 
in  commissions  ? 

4.  An  agent's  commission  for  selling  479  books  at  f3.50 
apiece  was  f  670.60.     What  was  the  rate  of  his  commission  ? 

5.  A  lawyer  procured  a  loan  for  an  improvement  company, 
charging  1-|  %  commission.  His  commission  was  14500.  What 
was  the  amount  of  the  loan  ? 

6.  A  dealer  in  typewriters  in  a  Western  city  sold  typewriters 
manufactured  in  New  York  State.  His  commission  was  35  %, 
out  of  which  he  paid  freight  charges  at  the  rate  of  14.50 
per  hundredweight. 

a.  If  the  weight  of  the  typewriters  averaged  50  pounds 
apiece  when  packed  for  shipment,  and  they  were  sold  at  an 
average  price  of  $103  each,  how  much  did  the  dealer  clear  on  a 
shipment  of  100  typewriters  ? 

h.  This  dealer  employed  an  agent,  paying  him  $10  a  week, 
and  20  %  commission.  The  agent  sold  two  typewriters  in  one 
week.     What  did  he  receive  for  his  week's  work  ? 

c.    How  much  did  the  dealer  gain  from  this  agent's  work  ? 

7.  An  agent  took  grocery  orders  on  a  commission  of  121%. 
He  sold  goods  amounting  to  $1352,  took  out  his  commission, 
paid  freight  charges  amounting  to  $30.75,  and  sent  the  remain- 
der of  his  collections  to  his  principal. 

a.    What  were  the  net  proceeds  of  the  sale  ? 
h.    How  many  dollars'  worth  of  goods  must  the  agent  sell 
to  earn  $568  in  commissions? 


COMMISSION  149 

8.  a.  An  agent  who  receives  115.  per  week  and  5J%  com- 
mission, must  sell  how  many  dollars'  worth  of  goods  in  a  year  to 
obtain  an  income  of  f  1566.50  ? 

h.  If  he  receives  no  compensation  but  his  commission,  what 
must  be  the  amount  of  sales  to  yield  him  the  same  income  as 
in  Question  a  ? 

9.  An  agent  who  had  charge  of  a  business  block  received  as 
his  commission  2  %  of  the  first  year's  rent  and  1  %  of  all  rents 
for  succeeding  years. 

a.  What  was  the  amount  of  his  commission  on  five-year 
leases  of  two  stores,  one  at  i  250  per  month  and  the  other  at 
1300  per  month? 

h.  What  was  his  commission  on  a  three-year  lease  of  an 
office  16  ft.  by  20  ft.,  the  annual  rent  being  at  the  rate  of  |  .80 
per  square  foot  of  floor  ? 

e.  His  commission  on  a  ten-year  lease  of  a  suite  of  banking 
rooms  was  |253.     What  was  the  annual  rent  ? 

10.  A  real  estate  agent  sold  my  property  in  Boston,  took  out 
his  commission  of  2  %,  and  remitted  to  me  the  remainder,  which 
was  $5880.     What  was  the  amount  of  his  commission  ? 

11.  A  commission  merchant  received  a  consignment  of  goods 
on  which  he  paid  182.50  freight  charges,  115.60  for  cartage, 
and  16  for  storage.  He  sold  the  goods,  deducted  his  Commis- 
sion of  8%,  and  his  disbursements  for  freight,  cartage,  and 
storage,  and  then  had  $7255.90  net  proceeds  of  the  sale,  which 
he  remitted  to  his  principal.  For  how  much  did  he  sell  the 
goods  ? 

12.  A  commission  merchant  sold  a  consignment  of  goods, 
paid  freight  charges  and  drayage  to  the  amount  of  $39.85, 
retained  his  commission  of  8%,  and  sent  the  remainder,  which 
was  $1685.15,  to  his  principal. 


150  GRAMMAR   SCHOOL   ARITHMETIC 

a.    What  was  the  amount  of  the  sales  ? 
h.    What  was  the  agent's  commission  ? 

13.  A  real  estate  agent  sold  a  tract  of  land,  and  bought  a 
business  block  with  the  money  received  for  the  land.  His 
commission  at  2  %  for  selling  and  |-  of  1  %  for  buying  amounted 
in  all  to  $1325.     For  how  much  did  he  sell  the  tract  of  land  ? 

14.  A  manufacturer  in  Pittsburg  sells  his  products  through 
a  commission  house  in  Philadelphia,  paying  8%  commission. 
What  is  the  selling  price  of  goods  for  which  the  manufacturer 
receives  16440  net  proceeds  ? 

15.  A  collector  receives  8^%  commission  on  all  the  money 
he  collects.  How  much  does  his  principal  receive  out  of  collec- 
tions for  which  the  collector  receives  $317.65  in  commissions? 

16.  A  farmer  sells  his  produce  through  a  commission  mer- 
chant in  the  city.  If  the  merchant's  commissions  average  9|  %, 
how  many  dollars'  worth  of  produce  must  the  farmer  sell  in 
order  to  receive  $1810  net  proceeds  ? 

COMMERCIAL  DISCOUNT 

27V.  It  is  customary  for  manufacturers,  wholesale  merchants, 
and  others  transacting  a  large  amount  of  business  to  distribute 
among  their  customers  printed  lists  of  the  articles  which  they 
offer  for  sale,  with  the  price  of  each  article.  These  lists  are 
called  price  lists.  The  goods  are  often  sold  at  a  lower  price 
than  that  given  in  the  price  list.  A  reduction  in  price  is  made 
sometimes  because  the  customer  buys  a  large  quantity  of  goods ; 
sometimes  because  other  dealers  are  selling  the  same  kind  of 
goods  at  a  lower  price  ;  sometimes  because  the  dealer  desires  to 
close  out  his  entire  stock  to  make  room  for  other  goods ;  some- 
times as  an  inducement  to  the  customer  to  pay  cash  instead  of 
paying  at  a  certain  time  after  the  purchase  of  the  goods.  Can 
you  mention  other  reasons  for  a  reduction  in  price  ? 


COMMERCIAL   DISCOUNT  151 

Two  or  more  reductions  are  often  made  in  the  price  of  the 
same  bill  of  goods,  as,  for  instance,  one  reduction  because  the 
market  price  of  that  kind  of  goods  has  fallen,  another  on  ac- 
count of  the  quantity  sold,  and  still  another  for  cash  payment. 

When  no  price  list  is  published,  goods  are  often  marked  at  a 
certain  price,  but  sold  at  a  reduction  from  that  price. 

278.  Tlie  marked  jprice,  or  the  pi^iee  given  in  a  price  list,  is 
called  the  list  price. 

279.  A  reduction  from  the  list  or  marked  price  of  goods  is  a 
commercial  discount  or  trade  discount. 

A  discount  for  cash  payment  is  sometimes  called  a  cash  discount.  A  dis- 
count because  of  the  quantity  of  goods  sold  is  sometimes  called  a  quantity 
discount. 

280.  The  sum  received  for  an  article,  after  all  discounts  have 
been  made,  is  the  net  price. 

281.  When  two  or  more  discounts  are  made  from  the  price  of  an 
article,  they  are  called  successive  discounts.  The  first  discount 
is  a  certain  per  cent  of  the  list  price,  the  second  a  certain  per 
cent  of  the  remainder,  the  third  a  certain  per  cent  of  the  sec- 
ond remainder,  and  so  on. 

282.  Oral 

1.  The  whole  of  anything  is  what  per  cent  of  it  ? 

2.  When  an  article  is  sold  at  a  discount  of  10  %  from  the 
list  price,  it  is  sold  for  what  per  cent  of  the  list  price  ?  When 
sold  at  a  discount  of  20  %  ? 

3.  I  bought  a  copy  of  Longfellow's  poems  listed  at  11.50, 
the  bookseller  allowing  me  20  %  discount.  How  much  did  I 
pay  ? 


152  GRAMMAR  SCHOOL   ARITHMETIC 

4.  I  can  buy  a  bicycle  for  ^40  and  pay  for  it  in  30  days,  or 
obtain  a  discount  of  2  %  by  paying  cash.  How  much  will  I 
save  by  paying  cash  ?     What  is  the  cash  price  ? 

5.  A  man  bought  a  bill  of  goods  at  10  %  discount.  He  paid 
8180  for  them. 

a.    What  per  cent  of  the  list  price  did  he  pay  ? 
h.   What  was  the  list  price  ? 

6.  By  paying  cash  for  a  bill  of  goods  I  obtained  a  discount 
of  2  %,  thereby  saving  $2.     What  was  the  amount  of  the  bill  ? 

7.  A  merchant  bought  from  a  jobber  goods  listed  at  $2000, 
receiving  a  discount  of  40  % .  What  was  the  entire  discount  ? 
What  did  he  pay  for  the  goods  ? 

8.  A  merchant  bought  a  bill  of  goods  at  a  discount  of  33 J  %. 
What  was  the  discount  on  goods  listed  at  190  ?  What  was  the 
net  price? 

9.  What  is  the  net  price  of  goods  listed  at  $  200  and  bought 
at  a  discount  of  30  %  ?     What  is  the  discount  ? 

10.  The  net  price  of  a  bill  of  goods  is  $12.  The  rate  of  dis- 
count is  40  % .     What  is  the  list  price  ?    What  is  the  discount  ? 

11.  The  net  price  of  a  bill  of  goods  is  f  30.  The  rate  of  dis- 
count is  40  % .     What  is  the  discount  ? 

12.  A  fruit  dealer  sold  me  ten  barrels  of  apples  at  f  2.50  a 
barrel.  They  arrived  in  poor  condition  and  he  discounted  the 
bill  20  (fo  •     How  much  did  I  pay  ? 

13.  A  discount  of  ^  is  equivalent  to  what  per  cent  discount  ? 
1  ?  1  ?  1  ?  1  ? 

6  •    ¥  •    ^  •    ¥  • 

283.  There  are  two  ways  of  treating  successive  discounts. 
For  example,  let  it  be  required  to  find  the  net  price  of  a  bill  of 


COMMERCIAL  DISCOUNT  153 

goods  listed  at  1 400,  on  which  successive  discounts  of  15%, 
10  %,  and  5  %  are  allowed. 

15  %  of  1400  =  160.     First  discount. 

f  400  -  $  60  =  1 340.    First  remainder, 
10  %  of  1 340  =  834.     Second  discount. 
$ 340  -  i  34  =  $  306.     Second  remainder. 
5  %  of  $306  =  $15.30.      Third  discount, 
$  306  - 15. 30  =  1 290. 70.     Net  price. 
Or 
The  net  price  is  95%  of  90%  of  85%  of  $400.     Find  the  net 
price  in  this  way  and  compare  results.     The  latter  method  is 
the  more  direct  and  in  most  cases  the  shorter. 

Written 

1.  Goods  listed  at  $3241  are  sold  at  a  discount  of  30%. 
What  is  the  net  price  ? 

Statement  of  Relation :  70%  of  $  3241  =  net  price. 

2.  A  man  bought  goods  at  15  %  discount.  What  was  the 
list  price  of  goods  that  cost  him  $59.50? 

Statement  of  Relation :  85%  of  the  list  price  =  $59.50. 
Which  term  of  relation  is  to  be  found? 

3.  A  merchant  saved  $4.50  by  paying  cash,  thus  obtaining  a 
discount  of  1|  %  on  a  bill  of  goods.  What  was  the  amount 
of  the  bill? 

Statement  of  Relation :  11%  of  the  amount  =  f  4.50. 
Which  term  of  the  relation  is  to  be  found? 

4.  Find  the  net  prices  of  the  following  bills  of  goods: 

List  Price  Discounts  List  Price  Discounts 

a.  $240       2%,  10%,    8%         d.  $312.50     10%,  10%,  10% 


I.  1300    10%, 

5%,    2% 

e.  1214 

2%,  10%,  20% 

e.  1870    80%, 

5%,    2% 

/.   1300 

15%,  10%,    5% 

154  GRAMMAR   SCHOOL   ARITHMETIC 

5.  A  bookseller  bought  books  at  an  average  discount  of 
38  %  from  the  list  price  and  sold  them  to  a  library  association 
at  an  average  discount  of  J  from  the  list  price.  How  much  did 
he  gain  on  a  bill  of  books  listed  at  $1S5? 

6.  A  druggist  sold  headache  powders  at  23  cents  a  box. 
They  were  listed  at  25  cents  a  box. 

a.  What  per  cent  discount  did  he  allow? 

b.  If  he  bought  them  at  40  %  discount,  what  did  he  pay  for 
seven  dozen  boxes? 

c.  What  per  cent  profit  did  he  make? 

7.  A  merchant  bought  carpet  at  60  cents  a  yard.  He  marked 
it  so  that  he  might  give  a  discount  of  10  %  and  still  make  20  %. 

a.  At  what  price  did  he  sell  the  carpet? 

b.  At  what  price  did  he  mark  it? 

8.  At  what  price  must  goods  costing  ^285  be  marked  so  that 
the  dealer  may  give  a  discount  of  5  %  and  still  make  a  profit  of 
18%? 

9.  A  merchant  sold  his  stock  of  goods  at  a  discount  of  10  % 
from  the  marked  price  and  still  made  a  profit  of  14  % . 

a.  If  he  received  $4560,  what  was  the  marked  price? 

b.  What  was  the  cost? 

10.  A  bill  of  goods  was  marked  at  45  %  above  cost,  and 
sold  at  a  discount  of  8^  %  from  the  marked  price.  The  marked 
price  was  $  725. 

a.  Find  the  cost. 

b.  Find  the  selling  price. 

11.  Steel  screws  are  listed  at  $8  a  great  gross,  and  succes- 
sive discounts  of  30  %,  40  %,  15%,  and  8  %  are  allowed.  What 
must  be  paid  for  40  great  gross? 

12.  Find  the  net  price  of  goods  listed  at  $720,  and  discounted 
at  5%,  10%,  and  20%. 


COMMERCIAL   DISCOUNT  155 

13.  A  speculator  bought  a  quantity  of  peaches  for  $280,  and 
marked  them  40  %  above  cost.  They  began  to  spoil  and  he  was 
obliged  to  sell  them  at  a  discount  of  40%  from  the  marked 
price.     Did  he  gain  or  lose,  and  how  much  ? 

14.  A  man  sold  two  vacant  lots  for  i960  apiece.  By  so 
doing  he  sold  one  at  a  discount  of  4  %  from  his  asking  price  and 
the  other  at  a  discount  of  20  %  from  his  asking  price.  Both 
were  marked  20%  above  cost. 

a.  What  did  each  cost  ? 

5.    What  was  his  entire  gain  ? 

15.  Two  merchants  have  the  same  kind  of  goods  marked  at 
the  same  price.  One  offers  discounts  of  25%,  20%,  and  5%. 
The  other  offers  discounts  of  5%,  20%,  and  25%.  Which  is 
the  better  offer  ? 

16.  Two  merchants  have  goods  exactly  alike,  listed  at  $200. 
One  offers  discounts  of  20%,  10%,  and  10%.  The  other 
offers  a  single  discount  of  37  % .  Which  is  the  better  offer,  and 
how  much  better  ? 

17.  A  carload  of  corn  containing  700  bushels  was  bought  on 
60  days'  time  at  48  cents  a  bushel.  The  purchaser  obtained  a 
discount  of  2|-  %  by  paying  cash.     What  did  the  corn  cost  him  ? 

18.  What  single  discount  is  equal  to  successive  discounts  of: 


a. 

10 

and  5  per  cent  ? 

i- 

25 

and  10  per  cent  ? 

b. 

^^ 

and  5  per  cent  ? 

h. 

25,  10, 

and  5  per  cent  ? 

c. 

15 

and  5  per  cent  ? 

I. 

30 

and  5  per  cent  ? 

d. 

15 

and  10  per  cent  ? 

m. 

,  30 

and  10  per  cent  ? 

e. 

16| 

and  10  per  cent  ? 

n. 

30,  10, 

and  5  per  cent  ? 

/. 

20 

and  5  per  cent  ? 

0. 

331 

and  5  per  cent  ? 

^• 

20 

and  10  per  cent  ? 

P- 

33X 

and  10  per  cent  ? 

h. 

20, 10,  and  5  per  cent  ? 

q- 

331  10, 

,  and  5  per  cent  ? 

i. 

25 

and  5  per  cent  ? 

r. 

40 

and  5  per  cent  ? 

156  GRAMMAR  SCHOOL  ARITHMETIC 

s.  40  and  10  per  cent  ?  w.  45  and  10  per  cent  ? 

t,   40,  10,  and    5  per  cent  ?  x.  50  and    5  per  cent  ? 

u,  40  and  20  per  cent  ?  ?/.  50  and  10  per  cent  ? 

v.  40,  20,  and    5  per  cent  ?  z.  50,  10,  and    5  per  cent  ? 

19.  $144  was  sufficient  to  pay  a  bill  on  which  discounts  of 
20%  and  10%  were  given.  What  was  the  amount  of  the  bill 
before  the  discounts  were  made  ? 

Statement  of  Relation:  90%  of  80%  of  the  amount  =  $144. 
When  the  product  of  three  factors  and  two  of  the  factors  are  given,  how 
may  the  remaining  factor  be  found  ? 

20.  What  is  the  list  price  of  a  bill  on  which  discounts  of 
10  %,  10  %,  and  5  %  make  the  net  price  1 153.90  ? 

21.  Two  successive  discounts  reduced  to  1108  the  price  of 
an  article  listed  at  $160.  One  of  the  discounts  was  25%. 
What  was  the  other  ? 

Statement  of  Relation : %  of  75%  of  $  160  =  $  108. 

When  the  product  of  three  factors  and  two  of  the  factors  are  given,  how 
may  the  remaining  factor  be  found?  That  factor  subtracted  from  100%  is 
the  required  discount. 

22.  What  discount,  in  addition  to  one  of  20%,  will  reduce  a 
price  from  $50  to  139.20? 

23.  What  list  price  will  give  a  net  price  of  $113.40  when 
discounts  of  30  %,  10  %,  and  10  %  are  made  ? 

24.  A  merchant  bought  goods  at  a  discount  of  35  %  from  the 
list  price  and  sold  them  at  a  discount  of  25  %  from  the  list  price. 

Hint. — Goods  listed  at  f  100  cost  him  $65  and  he  sold  them  for  $75. 
a.  What  was  his  profit  on  goods  listed  at  $350  ? 
h.   What  was  his  rate  per  cent  of  profit  ? 

c.  What  was  his  profit  on  goods  which  cost  him  $195  ? 

d.  What  was  the  list  price  of  goods  that  cost  the  merchant 
$1300? 


CONTRACTS  157 

25.  A  man  bought  goods  at  successive  discounts  of  25%, 
10%,  and  10%,  and  sold  them  at  successive  discounts  of  10% 
and  5  %  from  the  list  price. 

a.  What  was  his  gain  on  goods  listed  at  f  80  ? 
5.   His  gain  was  what  per  cent  of  the  cost  ? 

CONTRACTS 

284.  A  contract  is  an  agreement  between  two  or  more  parties 
for  doing  or  not  doing  a  particular  thing. 

In  making  a  contract  it  is  necessary  that  all  the  parties  agree 
to  the  same  thing.  For  instance,  in  bargains  for  the  purchase 
of  property,  if  the  seller  has  in  mind  one  piece  of  property, 
while  the  buyer  thinks  he  is  buying  a  different  piece  of 
property,  there  is  no  contract. 

It  is  generally  held,  also,  that  there  must  be  a  consideration. 
That  is,  when  one  party  makes  a  contract  with  another,  he  must 
pay,  or  agree  to  pay,  a  sum  of  money,  or  render  some  service,  or 
give  something  of  value,  in  return  for  what  he  receives  from  the 
other. 

There  are  many  kinds  of  contracts.  Among  the  commonest 
ones  are  the  following: 

Contracts  for  the  purchase  of  property. 

Contracts  for  the  rental  of  property. 

Contracts  for  the  payment  of  money  —  such  as  notes,  bonds, 
and  mortgages. 

Contracts  of  insurance. 

Contracts  of  employment  —  as  when  one  person  agrees  to 
work  for  another  for  a  certain  time  at  a  specified  salary. 


INSURANCE 

285.    Insurance  is  a  contract  whereby  one  party  (usually  an 
\surance  company^  agrees  to  pay  to  another  party  a  specified  sum 


158  GRAMMAR   SCHOOL   ARITHMETIC 

of  money  in  case  a  certain  event  shall  happen^  such  as  the  death  of 
some  person^  injury  to  the  person^  hy  accident^  destruction  of 
property  hy  fire  or  water ^  or  loss  of  property  hy  theft  or  accident. 

The  different  forms  of  insurance  are  known  as  life  insur- 
ance, accident  insurance,  fire  insurance,  marine  insurance,  etc., 
according  to  the  kind  of  risk  that  is  assumed  by  the  insurer. 

286.  The  written  or  printed  document  that  contains  the  terms  of 
an  insurance  contract  is  called  an  insurance  policy. 

287.  The  sum  which  the  insurer  agrees  to  pay  is  called  the 
face  of  the  policy. 

288.  The  sum  paid  hy  the  insured  to  the  insurer  is  called  the 
premium. 

Life  insurance  policies  are  in  force  for  a  term  of  years 
or  during  the  life  of  the  insured  ;  but  the  premium  is  usually 
paid  in  annual,  semi-annual,  or  quarterly  installments.  In- 
stallments after  the  first  are  called  renewals. 

Most  other  kinds  of  insurance  policies  are  for  a  shorter  time, 
and  the  premium  is  paid  in  one  sum  when  the  policy  is  issued. 

Accident  policies  are  usually  made  out  for  one  year,  though 
some  special  kinds,  like  railroad  accident  policies,  are  sold  for 
shorter  periods. 

Fire  insurance  policies  are  usually  for  three  years. 

The  premium  on  a  fire  insurance  policy  is  computed  at  a  cer- 
tain sum  for  each  $100  of  insurance,  or  a  certain  per  cent  of 
the  face  of  the  policy,  this  single  rate  covering  the  entire  time 
for  which  the  policy  is  given. 

The  premiums  on  life  insurance  policies  are  generally  com- 
puted at  a  certain  sum  for  each  |>1000  of  the  face  of  the  policy, 
the  sum  varying  according  to  the  age  of  the  insured  when  the 
policy  was  issued,  and  according  to  the  conditions  of  the 
contract. 


INSURANCE  159 

289.    The  following  forms  illustrate  some  kinds  of  insurance 
policies.     Only  the  essential  parts  of  each  contract  are  given. 

FIRE   INSURANCE  POLICY 
J\rn      258683  $2000 

THE 

MEGHANieS  INSaRANGE  GO/nPANY 

Incorporated  a.d.  1834.  Qp  B OSTEON 

En  Consitieration  of  tfje  .Stipulations  Ij^rein  nameti  anii  of 

80 

Twenty  Four  and  100 IBollars*  premium 

Does  Insure Jct^Qb  P.  Goettel for  the  term  of one  year 

from  the l.lUh .day  of Qctober \^04_^  at  noon, 

to  the IM^_ day  of 09to'ber_ I9_^«?_,  at  noon. 

against  all  direct  loss  or  damage  by  fire,  except  as  hereinafter  provided, 

To  an  amount  not  exceeding ^J^P^  Ab'PJ^§9Jl^^_ Dollars, 

to  the  following  described  property,  while  located  and  contained  as  described  herein, 

and  not  elsewhere,  to  wit: 

Jacob  P.  Goettel 

^AOyil_0Ti  the  three-  and  four-story  brick  building,  including  elevators  and 
all  attachments,  gas  and  water  pipes,  and  fixtures,  heating  ap- 
paratus and  fixtures,  and  plate  glass  in  doors  and  windows,  occu- 
pied for  storage  purposes,  situate  on  the  east  side  of  and  known  as 
Ho.  240  North  Salina  Street,  Syracuse,  NY.  Mechanic's  permit 
attached. 

Permission  given  for  the  use  of  gas,  kerosene  oil,  or  electric  lights 
on  said  building. 

Other  insurance  permitted  without  notice  until  required.    Light- 
ning clause  attached. 

*********** 

In  SJSitneSS  TOfjcreof,  this  Company  has  executed  and  attested  these  presents  this__  0^^^_ 

day  of. \lQt:9P§T- 19_(:^.        This  Policy  shall  not  be  valid  until  countersigned 

by  the  duly  authorized  Agent  of  the  Company  at P_yT9PJ^§^>.  ^-.  i  fC* 

Attest:    Jno.  A.  Snyder,  Secretary.               Samuel  Martin,  President. 
Countersigned  by fJ'SBLPJ.  ±  YJ'Ht Agent. 


160  GRAMMAR  SCHOOL  ARITHMETIC 


OEDINAET  LIFE  INSURANCE  POLICY 

THE  NORTH  STAR  MUTUAL  LIFE 
INSURANCE  COMPANY 

In  Consideration  of  the  application  for  this  Policy,  a  copy  of  which 
is  attached  hereto  and  made  a  part  hereof,  and  in  further  consideration  of  the 
payment  of 

100 
the  receipt  whereof  is  hereby  acknowledged,  and  of  the      Annual       payment 

of  a  like  sum  to  the  said  Company,  on  or  before  the J^trst day  of 

3anuarg         in  every  year  during  the  continuance  of  this  Policy,  promises 

to  pay  at  its  office  in  Milwaukee,  Wisconsin,  unto jlHarg  Boe 

" ,  Beneficiar_£_, 

^ aggifc  of  3fol}n  mot  the  Insured,  of 


Bes  ilHomcs  in  the  State  of gofoa 

subject  to  tfje  rtgfjt  of  tlje  gnsurctt,  fjerebg  rcserbctt,  to  cfjange  tfje  Beneficiarg  or 
beneficiaries  the  sum  of        ^en  gTl^ousanti  —        Dollars^ 

upon  receipt  and  approval  of  proof  of  the  death  of  said  Insured  while  this 
PoHcy  is  in  full  force,  the  balance  of  the  year's  premium,  if  any,  and  any 
other  indebtedness  on  account  of  this  Policy  being  first  deducted  therefrom  ; 
provided,  however,  that  if  no  Beneficiary  shall  survive  the  said  Insured,  then 
such  payment  shall  be  made  to  the  executors,  administrators  or  assigns  of  the 
said  Insured. 

In  Witness  Whereof,  THE  NORTH  STAR  MUTUAL  LIFE  INSURANCE  COM- 
PANY, at  its  office  in  Milwaukee,  Wisconsin,  has  by  its  President  and  Secretary,  executed 
this  contract,  this Firsf ^y  pf  January  one  thousand  nine  hundred  and 

eight. 

S.  A.  Hawkins^  Secretary.  Z.  H.  Perkins^  President. 


INSURANCE  161 

ACCIDENT   INSURANCE   POLICY 

dtiited  Casualty  Company 

Ktt  Consitieration  of  — Twenty-five Dollars'  premium  and  the  warranties 

and  agreements  in  the  application  for  this  policy,  which  application  is  hereby- 
made  a  part  hereof,  the  United  Casualty  Company,  herein  called  the  Com- 
pany, insures,  subject  to  the  provisions,  conditions,  definitions  and  limits  herein, 

John  Doe of Harrisburg,  Pa 

by  occupation : Printer 

herein  called  the  Insured,  for twelve months,  beginning  at  noon,  standard 

time,  on  the first day   of January, 190-5_,  against    loss  as 

herein  provided  caused  by  bodily  injury  effected  exclusively  and  directly  by  ex- 
ternal, violent  and  accidental  means  which,  independently  of  any  and  all  other 
causes,  immediately,  wholly  and  continuously  disables  him,  to  wit: 

I     A    LOSS  OF  LIFE, Ten  Thousand DoWurs  ($  .10,000)  ; 

B    LOSS  OF  BOTH  EYES,  the  amount  stipulated  for  loss  of  life  ; 

C    LOSS  OF  BOTH  HANDS,  the  amount  stipulated  for  loss  of  life  ; 

D    LOSS  OF  BOTH  FEET,  the  amount  stipulated  for  loss  of  life  ; 

E    LOSS  OF  ONE  HAND  and  ONE  FOOT,  the  amount  stipulated  for  loss  of  life  ; 

F    LOSS  OF  ONE  ARM,  three-fifths  the  amount  stipulated  for  loss  of  life  ; 

G    LOSS  OF  ONE  LEG,  three-fifths  the  amount  stipulated  for  loss  of  life  ; 

H    LOSS  OF  ONE  HAND,  one-half  the  amount  stipulated  for  loss  of  life  ; 

I     LOSS  OF  ONE  FOOT,  one-half  the  amount  stipulated  for  loss  of  life  ; 

J     LOSS  OF  ONE  EYE,  one-quarter  the  amount  stipulated  for  loss  of  life  ; 

K    TOTAL  LOSS  OF  TIME, Twenty-five Dollars  per  week,  not  to  exceed 

104  consecutive  weeks ; 
L    PARTIAL  LOSS  OF  TIME,  one-half  the  amount  stipulated  for  total  loss  of  time  per 

week,  not  to  exceed  30  consecutive  weeks. 
*********** 

3En  SjJitnegg  fafjercof  the  SEniteti  ffiasualtg  CTompang  has  caused  this  policy  to  be  signed  by  its 
President  and  Secretary,  but  it  shall  not  be  in  force  until  countersigned  by  a  duly  authorized  rep- 
resentative of  the  Company. 

Richard  Johnson,    President.  Dan  J.    Seward,    secretary. 

Countersigned 

Herbert  W.  Greenland,  Agt. 

290.    Written 

1.  a,  A  wooden  dwelling  house  in  a  city  was  insured  for 
three  years  for  f  3500,  the  rate  of  premium  being  $.Qb  on  |100 
of  insurance  for  three  years.     Find  the  premium. 


162  GRAMMAR  SCHOOL  ARITHMETIC 

h.  How  much  did  the  owner  pay  in  premiums  in  twelve 
years,  at  this  rate  ? 

e.  The  rate  of  premium  for  brick  dwellings  in  the  same  city 
is  bb^  on  ilOO,  for  three-year  policies.  Find  the  premium  for 
$4200  of  insurance  on  a  brick  dwelling  in  that  city. 

d.  The  insurance  agent  who  wrote  the  policy  in  question  c 
received  as  his  commission  25%  of  the  premium.  Find  the 
agent's  commission. 

2.  A  schoolhouse  in  a  Western  city  is  insured  for  three 
years  for  128,000,  at  |%,  The  agent's  commission  is  20%  of 
the  premium.     Find  the  agent's  commission. 

3.  A  farmer  in  Pennsylvania  has  his  house  insured  for 
$900  and  his  barns  for  $1350,  the  premium  for  the  former 
being  |%  and  for  the  latter  1^  %,  for  three-year  policies. 
What  is  the  farmer's  annual  expense  for  insurance  ? 

4.  a.  The  premium  for  insuring  a  mill,  in  a  small  village, 
for  $2000,  amounted  to  $75  a  year.  What  was  the  annual 
rate  of  premium  ? 

Statement  of  Relation:  %  of  $2000  =  |75. 

Which  term  of  relation  is  to  be  found  ? 

h.  What  was  received  by  the  agent  who  wrote  three  annual 
policies  on  this  mill,  his  commission  being  15  %  of  the  premiums'^ 

5.  A  merchant's  stock  of  goods  is  insured  for  -|  of  its  value, 
for  three  years,  at  |^%.  If  the  stock  is  worth  $7500,  what  is 
the  annual  expense  for  insurance  ? 

6.  I  pay  $28.50  for  three  years'  insurance,  the  rate  of 
premium  being  |  %.     How  much  insurance  have  I  ? 

Statement  of  Relation:  f  %  of  $ =  $28.50. 

7.  The  premium  for  insuring  my  house,  at  70^  per  $100,  is 
$38.50.     What  is  the  face  of  the  policy  ? 


INSURANCE  163 

8.  A  machine  shop  is  insured  for  three  years  at  a  cost  of 
$114.     If  the  rate  is  1|%,  what  is  the  face  of  the  policy  ? 

9.  An  agent  received  $5,25  as  his  commission  for  insuring 
a  house  for  ^  of  its  value.  The  rate  of  premium  was  ^  %  and 
the  agent  received  25  %  of  the  premium, 

a.  What  was  the  premium  ? 

b.  What  was  the  face  of  the  policy  ? 

c.  What  was  the  value  of  the  house  ? 

10.  How  many  dollars  of  insurance  must  an  agent  secure  in 
order  that  he  may  obtain  $46,35,  if  his  commission  is  15%  of 
the  premiums  and  the  premiums  are  1-|-  %  of  the  insurance  ? 

11.  A  man  had  an  accident  insurance  policy  which  cost  him 
$25  a  year.  After  he  had  paid  three  years'  premiums,  he  was 
injured  by  an  accident  and  received  $20  a  week  for  six  weekso 

a.  The  man  received  how  much  more  than  he  paid  ? 

5.  If  the  agent  received  30  %  of  the  premiums,  how  much 
did  the  insurance  company  lose  by  insuring  this  man  ? 

c.  If  the  company  insured  ten  other  men  for  the  same  time 
at  the  same  rate,  and  none  of  them  made  any  claim  for  injuries, 
how  much  more  did  the  company  receive  from  the  eleven  men 
than  it  paid  out  on  account  of  the  one  man's  injuries  ? 

12.  A  house  worth  $  3600  was  insured  for  |-  of  its  value,  and 
the  contents,  worth  $2800,  were  insured  for  1  of  their  value. 
The  rate  of  insurance  was  65/  on  $100.  The  house  and  con- 
tents were  entirely  destroyed  within  a  year. 

a.  What  did  the  company  lose  by  insuring  the  property  ? 

5.  What  did  the  owner  lose  by  the  fire? 

c.  What  did  the  owner  gain  by  having  the  property  insured  ? 

13.  A  mill  owner  had  his  mill  insured  every  year  by  one 
company,  for  $23,000,  at  ^%.     After  he  had  paid  five  annual 


164  GRAMMAR  SCHOOL  ARITHMETIC 

premiums,  the  mill  was  damaged  by  fire  to  the  amount  of 
$3150,  which  was  paid  in  full  by  the  insurance  company. 

a.   How  much  did  the  owner  gain  by  having  the  mill  insured  ? 

h.   How  much  did  the  company  lose  by  insuring  the  mill  ? 

14.  Property  worth  $48,600  is  insured  for  |  of  its  value  at  a 
cost  of  $364.50.     What  is  the  rate  ? 

15.  a.  Turn  to  the  life  insurance  policy  on  page  160.  If 
John  Doe  lives  to  be  seventy  years  old,  and  pays  35  re- 
newals, besides  the  first  premium,  how  much  money  will  he 
have  paid  to  the  insurance  company  ? 

h.  How  much  would  the  first  premium  amount  to  in  35 
years,  if  put  on  interest  at  the  rate  of  5  %  per  year,  simple 
interest  ? 

c.  If  the  agent  through  whom  John  Doe  obtained  this  policy 
received  as  his  commission  40  %  of  the  first  premium  and  5  % 
of  all  renewals  for  the  first  ten  years,  how  much  did  he  receive 
in  all  ? 

•  16.  The  premium  on  Mr.  Wilson's  accident  policy  was  at  the 
rate  of  $5.00  per  $1000  of  the  face  of  the  policy.  The  agent's 
commission  was  25%  of  the  premium.  If  the  agent  received 
$6.25,  what  was  the  face  of  the  policy  ? 

17.  A  man  took  out  a  $5000  ten-year  endowment  life  insur- 
ance policy,  on  which  the  semi-annual  premium  was  $55.95  per 
$1000  of  the  face  of  the  policy.  If  he  lived  ten  years  after 
taking  the  policy,  and  paid  all  the  premiums  when  due,  how 
much  did  he  pay  to  the  company  ? 

18.  A  merchant  has  his  stock  of  goods  insured  for  $18,000  at 
2J%  for  three  years,  his  building  for  $12,500  at  2%  for  three 
years,  one  boiler  at  $15  a  year,  an  elevator  at  $35  a  year,  and 
plate  glass  at  $12  per  year,  with  30%  off,  on  the  plate  glass. 
How  much  does  his  insurance  cost  in  three  years  ? 


INTEREST  165 

19.  A  man  starting  on  a  journey  bought  a  12-day  accident 
policy  at  f2.50.  When  that  expired  he  bought  a  10-day 
policy  at  ^2.00,  and  after  that  a  7-day  policy  at  25  cents  a 
day.  How  much  would  he  have  saved  by  investing  at  first  in 
a  30-day  policy  that  cost  14.50? 

INTEREST 

291 .  Money  paid  for  the  use  of  money  is  interest. 

292.  Money  for  the  use  of  which  interest  is  paid  is  the  princi- 
pal. 

293.  The  sum  of  the  principal  and  interest  is  the  amount. 

294.  The  sum  to  he  paid  for  the  use  of  money  is  always  deter- 
mined by  taking  a  certain  per  cent  of  the  principal. 

295.  The  number  of  hundredths  of  the  priiicipal  taken  as  the 
interest  for  one  year  is  the  rate  of  interest.  For  instance,  if  a 
sum  of  money  is  borrowed  and  6  %  of  that  sum  is  the  interest 
for  one  year,  the  rate  of  interest  is  6  % . 

296.  The  rate  of  interest  which  is  fixed  by  law  is  called  the 
legal  rate.  In  a  majority  of  the  states  the  legal  rate  is  6%. 
In  some  states  it  is  greater  than  6  %,  and  in  some  states  less. 

A  lower  rate  than  the  legal  rate  is  always  allowed  by  law  if  the  debtor 
and  creditor  so  agree.  In  some  states  a  higher  rate  than  the  legal  rate  is 
allowed,  if  the  debtor  and  creditor  so  agree ;  in  others  a  higher  rate  than  the 
legal  rate  is  forbidden  by  law.     What  is  the  legal  rate  where  you  live  ? 

297.  Interest  at  a  higher  rate  than  that  permitted  by  law  is 
usury. 

298.  Oral 

1.  Mr.  Smith  borrowed  $100  from  Mr.  Arnold  for  1  yr.  At 
the  end  of  the  year  Mr.  Smith  repaid  the  money  which  he  had 


166  GRAMMAR  SCHOOL  ARITHMETIC 

borrowed  and  also  paid  Mr.  Arnold  6  %  interest.  How  much 
was  the  interest  ?  What  was  the  principal  ?  How  much  did 
Mr.  Arnold  receive  in  all  ?  What  is  this  sum  called  ?  Who 
was  the  debtor  ?     Who  was  the  creditor  ? 

2.  What  is  the  interest  on  1 500  at  5  %  for  1  yr.  ?  On  $  800  ? 
On  1 900  ?     On  $  300  ?     On  1 1000  ?     On  1 250  ? 

3.  What  is  the  interest  on  1 500  for  1  yr.  at  5  %  ?  At  10  %  ? 
At  7  %  ?     At  4  %  ?     At  3  %  ?     At  8  %  ? 

4.  What  is  the  interest  on  $  1000  at  5  %  for  1  yr.  ?  For 
2yr.  ?     ForSyr.?     For  8  yr.  ?     ForlOyr.? 

5.  What  is  the  interest  on  1 100  for  2  yr.  at  6  %  ?  At  4  %  ? 
At  3  %  ?     At  9  %  ?    At  8  %  ? 

6.  Six  months  are  what  part  of  a  year  ?  3  mo.  ?  4  mo.  ? 
8  mo.  ?     9  mo.  ?     10  mo.  ?     1  mo.  ? 

7.  What  is  the  interest  on  1 600  for  1  yr.  at  6  %  ?  For 
6  mo.  ?     For  3  mo.  ?     For  4  mo.  ?     For  8  mo.  ?     For  9  mo.  ? 

8.  Make  and  solve  many  problems  similar  to  the  above, 
multiplying  the  interest  for  one  year  by  the  number  of  years, 
treating  months  as  fractions  of  a  year. 

299.    Written 

1.    a.    What  is  the  interest  on  1800  for  2  yr.  6  mo.  at  5  %  ? 

How  do  we  find  the  interest  for  one  year  ?    For  2i  years? 
b.    What  is  the  interest  on  $840  for  1  yr.  9  mo.  at  4 J  %? 
105 

m 

X  ^x  7  =  166.15    Ans. 
100 


INTEREST  167 

9  4- 

How  does  —  compare  in  value  with  -— ^  ? 
200  100 

We  multiply  4-1  and  100  by  2  to  obtain  2^^. 
1  yr.  9  mo.  =  how  many  months? 

2.  Find  the  interest  on 

a.  1750  for  2  yr.  at  6%. 

h.  1 375  for  1  yr.  6  mo.  at  6  %. 

c.  $500  for  2yr.  at  3|%. 

d.  f  625  for  6  mo.  at  4  %.  (6  mo.  =  what  part  of  a  year  ?) 

e.  I  342.40  for  1  yr.  3  mo.  at  4J  %. 
/.  1 279.75  for  1  yr.  2  mo.  at  6 %. 
g.  $364.50  for  2  yr.  8  mo.  at  ^%. 
h.  %  640  for  1  yr.  9  mo.  at  51  %. 

300.    Oral 

1.  In  computing  interest,  one  year  is  assumed  to  be  360  days, 
and  one  month  30  days.  On  that  assumption,  1  day  is  what 
fraction  of  a  year?  2  days  ?  3  days?  7  days?  245  days  ? 
430  days  ?     83  days  ?     792  days?     879  days ?     90  days? 

2.  The  interest  on  a  sum  of  money  for  1  day  is  what  part 
of  the  interest  for  a  year  ? 

3.  The  interest  on  a  sum  of  money  for  9  days  is  what  part 
of  the  interest  for  a  year  ?  The  interest  for  231  days  is  what 
part  of  the  interest  for  a  year? 

4.  1  yr.  and  15  da.  are  how  many  days,  counting  360  days 
as  a  year? 

5.  The  interest  on  a  sum  of  money  for  1  yr.  15  da.  is  how 
many  360ths  of  the  interest  for  a  year  ? 

6.  The  interest  on  a  sum  of  money  for  3  mo.  20  da.  is  how 
many  360ths  of  the  interest  for  one  year  ? 

7.  The  interest  for  5  mo.  10  da.  is  how  many  360ths  of  the 
interest  for  one  year? 


168  GRAMMAR  SCHOOL  ARITHMETIC 

301.  Written 

1.  Find  the  interest  on  $240  for  1  yr.  3  mo.  18  da.  at  ^%. 

3 

^  52 

1  yr.  =  360  da.        |^     _7_  ^0^  _ 
3  mo.  =    90  da.  i     ""  ^ff^p""  ^  "^^^'^"^  "^'''• 

18  da.  100       ^ 

1  yr.  3  mo.  18  da.  =  468  da.,  or  ^^^  yr. 

2.  Find  the  interest  on 

a.  1 700  for  30  da.  at  6%. 

h.  $450  for  45  da.  at  5%. 

c.  11380  for  82  da.  at  4-1%. 

d.  $3000  for  2  mo.  20  da.  at  7  %. 

e.  $6540  for  1  yr.  15  da.  at  5%. 

/.  $2700  for  1  yr.  2  mo.  12  da.  at  4  %. 

ff.  $450  for  1  yr.  6  mo.  6  da.  at  5|  %. 

h.  $280  for  2  yr.  2  mo.  17  da.  at  3  %. 

i.  $519.16  for  173  da.  at  5%. 

y.  $249.83  for  1  yr.  5  mo.  14  da.  at  6%. 

k.  $931  for  1  yr.  11  mo.  19  da.  at  21%. 

I.  $67,000  for  2  yr.  17  da.  at  3%. 

m.  $864.13  for  9  mo.  16  da.  at  4^%. 

n.  $4182  for  1  yr.  4  mo.  11  da.  at  8%. 

0.  $180.55  for  10  mo.  23  da.  at  6^  %. 

302.  Oral 

1.  Find  the  interest  on 

a.    $300  for  1  yr.  at  31%.  e.    $300  for  30  da.  at  4  % 

h.    $200  for  2  yr.  at  51%.  d.    $700  for  6  mo.  at  3%. 

2.  From  Jan.  1,  1908,  to  June  1,  1908,  is  what  part  of  a 
year  ?    Find  the  interest  on  $  2500  at  6  %  for  that  time. 


INTEREST  169 

3.  Find  the  interest  on  f  200  from  July  1, 1908,  to  Jan.  1, 1909, 
at  the  legal  rate  where  you  live. 

4.  What  is  the  interest  on  $400  from  Nov.  1, 1907,  to  Nov.  1, 
1909,  at  the  legal  rate  in  your  state  ? 

5.  If  I  borrow  $250  on  the  first  of  January  and  pay  the  debt 
on  the  first  of  the  following  January,  with  interest  at  the  legal 
rate  in  your  state,  how  much  do  I  pay  ? 

303.     Written 

1.  What  is  the  amount  of  |700  when  put  at  interest  at  5% 
from  Nov.  21,  190T,  to  June  3,  1909? 

1909  yr.  6  mo.  3  da. 

1907 11 21 

1  yr.  6  mo.         12  da.     Biff,  in  Time 

7  46  $700.00 

ZM  X  -f-  X  il  =  153.67  Interest  ^3.67 

1        ^^     ?W  $15S,67  Amount,  Ans. 

6 

2.  Find  the  amount  of 

a.  1250  from  April  7,  1905,  to  Oct.  19,  1906,  at  6%. 

b.  $5000  from  Sept.  15,  1905,  to  May  21,  1907,  at  6%. 

c.  $348  from  July  25,  1902,  to  March  11,  1904,  at  5%. 

d.  $1000  from  Jan.  28,  1907,  to  Jan.  21,  1909,  at  5-1  %. 

e.  $875  from  Sept.  30,  1908,  to  Feb.  24,  1909,  at  4i  %. 
/.    $3980  from  March  2,  1901,  to  July  2,  1903,  at  41%. 
g.   $600  from  Oct.  12,  1899,  to  April  12,  1901,  at  7%. 
h.    $1350  from  Aug.  25,  1907,  to  Dec.  5,  1908,  at  51%. 
^.    $163.50  from  Dec.  16,  1907,  to  Jan.  1,  1909,  at  8%. 

3.  Mr.  Anderson  borrowed  $700  May  15,  1907,  and  agreed 
to  pay  it  June  3,  1908,  with  6  %  interest.  How  much  did  he 
have  to  pay? 


The  money  is  on  interest  for 


170  GRAMMAR  SCHOOL  ARITHMETIC 

4.  What  is  the  amount  of  1600  when  put  at  interest  from 
June  24, 1904,  to  May  19, 1906,  at  the  legal  rate  where  you  live? 

5.  Compute  the  interest  on  $428.70  from  Oct.  18,  1908,  to 
July  13,  1910,  at  the  legal  rate  where  you  live. 

INTEREST  FOR  SHORT   PERIODS 

304.    When  money  is  on  interest  for  less  than  a  year,  it  is 
customary  to  compute  the  time  in  days. 

1.  What  is  the  interest  on  11575.25  from  Jan.  9,  1904,  to 
March  15,  1904,  at  3  %  ? 

'22  da.  left  in  Jan. 
29  da.  in  Feb. 
15^  da.  in  March 
66  da.  Term  of  Interest 

2.  Compute  the  interest  on 

a.  1600  from  April  21  to  Aug.  3,  at  7%. 

5.  1845.60  from  Sept.  1  to  Dec.  24,  at  6  %. 

c.  $570  from  April  25  to  Aug.  13,  at  5|  %. 

d.  1473.70  from  June  1  to  July  31,  at  8  %. 

e.  $1857  from  Nov.  30  to  Dec.  31,  at  7%. 

3.  Compute  the  interest  on 

a.    1900  from  Dec.  18,  1903,  to  Feb.  21,  1904,  at  6|-%. 
5.    $388.20  from  Dec.  18,  1906,  to  Feb.  21,  1907,  at  6%. 

c.  $1880  from  Dec.  19,  1905,  to  March  1,  1906,  at  3i%. 

d.  $1230  from  Dec.  19,  1907,  to  March  1,  1908,  at  6%. 

e.  $870  from  Nov.  1,  1908,  to  April  1,  1909,  at  6%. 

4.  Compute  the  amount  of 

a.  $496  from  June  15  to  Oct.  15,  1901,  at  5%. 

b.  14000  from  Dec.  1,  1903,  to  Feb.  1,  1904,  at  6%. 

c.  $460.80  from  May  8  to  July  7,  1908,  at  5  %. 

d.  $500  from  Sept.  30,  1905,  to  Feb.  10,  1906,  at  6  %. 


INTEREST  171 

5.  On  the  first  day  of  May,  1907,  Mr.  Blank  borrowed 
f  1800  with  which  to  buy  an  automobile,  agreeing  to  pay  the 
money  with  interest  at  6  %  on  the  10th  day  of  September.  He 
sold  the  automobile  for  fl350  on  the  10th  of  September. 
How  much  money  must  he  put  with  what  he  received,  in  order 
to  pay  his  debt  ? 

EXACT  INTEREST 

305.  When  a  day  is  called  3  J^  of  a  year,  in  computing  inter- 
est, the  interest  obtained  is  a  trifle  greater  than  it  would  be  if 
each  day  were  taken  as  ^J^  of  a  year — its  exact  value.  Inter- 
est computed  by  the  usual  method  is  therefore  slightly  inexact; 
yet  business  men  seem  to  consider  that  its  greater  convenience 
compensates  for  its  lack  of  accuracy. 

306.  Exact  interest  is  interest  computed  hy  taking  as  many 
^Qbths  of  the  interest  on  the  given  principal  for  one  year  as  there 
are  days  in  the  interest  period. 

The  exact  method  of  computing  interest  is  employed  by  the  United  States 
government  and,  to  a  limited  extent,  elsewhere. 

The  process  is  the  same  as  that  given  in  the  preceding  pages,  except  that 
the  last  factor  has  365,  instead  of  360,  for  its  denominator. 

307.  Written 

1.    What  is  the  exact  interest  on  $731.46,  at  8%,  from  Jan. 


29  to  July  22,  1908  ? 

Jan.     2  da. 

Feb.  29  da. 

Mar.  31  da. 

10.02                   35 

Int.  Period  - 

Apr.  30  da. 

mU-n..    8    an 

May  31  da. 

1         ^  100  '^  ^^ 

June  30  da. 

n 

July  22  da. 

T 

otal,  175  da. 

128.056  or 
128.06  Am, 


172  GRAMMAR  SCHOOL   ARITHMETIC 

2.  Find  the  exact  interest  on 

a.  $5000  at  5%  from  Oct.  5,  1905,  to  April  3,  1906. 

h,  $584  at  4%  from  Jan.  7  to  May  5,  1908. 

c.  $109.50  at  3%  from  May  5  to  Sept.  6,  1905. 

d.  $2190  at  7%  from  Nov.  15,  1908,  to  April  1,  1909. 

e.  $75.50  at  31%  for  90  da. 

3.  A  man  borrowed  $500  on  the  5th  of  May.     How  much 
is  due  on  the  debt  July  first,  computing  exact  interest  at  5  %  ? 

4.  What  is  the  difference  between  the  common  and  the  exact 
interest  at  5  %  on  $525,600  for  15  da.  ? 

5.  What  is  the  amount  of  $328.50,  computing  exact  interest 
at  7  %,  from  June  12  to  Aug.  28  ? 

6.  Find  the  amount  of  $1095  for  146  da.,  computing  exact 
interest  at  5^  % . 

7.  Find  the  exact  interest  on  $8760,  at  41%,  from  Oct.  15, 
1908,  to  Feb.  15,  1909. 

PROBLEMS  IN  INTEREST 
308.    Oral 

1.  In  the  preceding  examples  in  interest  we  have  found  in 
every  case  that  the  interest  is  the  product  of  what  factors  ? 

2.  When  we  have  given  any  number  of  factors,  what  must 
we  do  to  find  the  product  ? 

3.  When  we  have  given  the  product  of  two  factors,  and  one 
of  the  factors,  how  may  we  find  the  other  factor  ? 

4.  Which  term  in  division  is  always  a  product?      Which 
terms  are  factors? 

5.  Which  term  in  multiplication  is  the  product?      Which 
terms  are  factors? 


INTEREST 


173 


6.  8x7 
found  ? 

7.  8x? 

found  ? 


?  Which  terms  are  given  ?  Which  is  to  be 
36.  Which  terms  are  given?  Which  is  to  be 
Which  terms  are  given?     Which  is  to  be 


8.  ?  X  7  =  56 
found  ? 

9.  When  we  have  given  the  product  of  three  factors,  and 
two  of  the  factors,  how  may  we  find  the  remaining  factor  ? 

10.    5x3x2  =  ?     Which  terms  are  given  ?     Which  is  to  be 
found  ? 

30.     Which  terms  are  given  ?     Which  is  to 


11.    5  x  3  X  ? 
be  found  ? 


30. 


12.  5x?x2 
be  found? 

13.  ?  X  3  X  2  =  30. 

be  found? 


Which  terms  are  given? 
Which  tferms  are  given? 


Which  is  to 


Which  is  to 


14.  In  each  of  the  following  examples,  tell  which  terms  are 
given,  arid  which  is  to  be  found,  and  find  the  term  which  is 
wanting. 


a. 

3x7x2=? 

h. 

?x7x2  =  42 

c. 

3  X  ?  X  2  =  42 

d. 

3x7x?  =  42 

e. 

7x5x2=? 

/. 

8x?x3  =  48 

9- 

9x6x?  =  108 

h. 

4  X  7  X  ?  =  112 

i. 

lOx    ?  X 

10  =  10,000 

J- 

3xllx 

?  =  99 

k. 

6x    ?  X 

5  =  120 

I 

7x   2  X 

?  =  700 

m. 

?xl2  X 

5  =  600 

n. 

?xl3  X 

4  =  104 

0. 

5  X    5  X 

?  =  125 

P- 

?x    6  X 

7  =  210 

15.    When  we  have  given  the  principal,  rate,  and  time  ex- 
pressed in  years,  how  is  the  interest  found? 


174  GRAMMAR   SCHOOL   ARITHMETIC 

16.  The  principal,  rate,  and  time  expressed  in  years  are  what 
of  the  interest  ? 

17.  When  the  principal,  rate,  and  interest  are  given,  how 
may  the  time  be  found? 

18.  When  the  principal,  time,  and  interest  are  given,  how 
may  the  rate  be  found  ? 

19.  When  the  rate,  time,  and  interest  are  given,  how  may 
the  principal  be  found? 

309.    Written 

1.  The  interest  on  1720  for  1  yr.  8  mo.  11  da.  is  $61.10. 
Find  the  rate. 

Statement  of  Relation :  ?^  x  Rate  x^  =  $61.10. 
1  360 

Which  terms  of  relation  are  given?    Which  is  to  be  found?    How  shall 
we  find  it? 

4 

Solution 
2 

^x  611  =  1222. 

1    m 

Rate  =  61.10  -  (—  x  ^]  =  61.10  -  1222  =  .05,  or  5%  Atis. 

2.  At  what  rate  of  interest  will  $2350  gain  $94  in  8  mo.  ? 

3.  When  the  interest  on   |240  for  1  yr.  7  mo.  is  $30.40, 
what  is  the  rate? 

4.  At  what  rate  will  $1600  amount  to  $1718.60  in  1  yr. 
7  mo.  23  da.  ? 

Statement  of  Relation:  ^^^  x  Rate  x  —  =  $118.60. 
-^  1  360 

What  is  $118.60?     How  is  it  obtained? 

5.  At  what  rate  will  $52.50  double  itself  in  16  yr.  8  mo.  ? 


INTEREST  ,  175 

6.  At  what  rate  will  any  sum  double  itself  in  14  yr.  ? 

7.  At  what  rate  will  any  sum  double  itself  in  16  yr.  8  mo.  ? 

8.  At  what  rate  must  f  960  be  put  at  interest  to  gain  199.20 
in  1  yr.  3  mo.  15  da.  ? 

9.  Interest  1110.72,  principal  $3460,  time  8  mo.  16  da.  Find 
the  rate. 

310.    Written 

1.  In  what  time  will  $5000  gain  $375  if  put  at  interest 
at  41%? 

Statement  of  Relation:  i^^  X  ii  X  | .  ^'"^^   \  =  |375.     ^ 
•^  1  100      lin  years  J 

Which  terms  of  relation  are  given  ? 
Which  term  is  to  be  found  ? 

Solution 

1      m 

Time  =  E5  ^  (5000  ^  Jt£\      375  ^  225  =  If  yr. 

1      V  1      100  /  '  "^ 

or,   1  yr.  8  mo.  Ans. 

2.  For  what  time  will  $101.50  pay  the  interest  on  $725 
at  7  %  ? 

3.  A  young  man  borrowed  $3000  from  his  father,  paying 
him  41  %  interest  every  year.  How  long  must  the  father 
permit  the  debt  to  run  in  order  to  receive  $  945  in  interest  ? 

4.  In  what  time  will  $4816  on  interest  at  ^%  earn 
$421.40? 

5.  In  what  time  will  $1200  earn  $306  if  put  on  interest  at 


176  GRAMMAR  SCHOOL  ARITHMETIC 

6.  In  what  time  will  1210  bear  $25.62  interest,  at  9% 
per  annum  ? 

Statement  of  Relation:  $^  x  —  x  I   '^^"^^    1  =  $25.62. 
-^  1         100      1  in  years/ 

Solving  as  indicated  above,  the  time  is  1\^  yr. 

4 

H  yr.  =  ^  X  ^mo.  =  ff  mo.  =  4^  mo. 

^P       1 
15 

2 

A  mo.  =  —  X  ^  da.  =  8  da.  1  yr.  4  mo.  8  da.    Ans. 

JL^       1 

7.  $217  will  pay,  the  interest  on  82000  for  how  long  at 
6%? 

8.  For  what  time  will  $25.62  pay  the  interest  on  $210* 
at9%? 

9.  In  what  time  will  $231  put  at  interest  at  5%  amount  to 
$243. 70  J? 

10.  1630  will  pay  the  interest  on  $3500  at  5%  for  what 
time  ? 

11.  In  what  time  will  $810  amount  to  $823.23  if  put  at 
interest  at  7  %  ? 

12.  In  what  time  will  $1896  amount  to  $2006.60  at  5%  ? 

13.  A  note  for  $1800  with  interest  at  6%  amounted  to 
$1828.50  when  it  was  paid.     How  long  had  the  note  run  ? 

14.  A  man  borrowed  $1280  at  4|%  interest  and  paid  the 
debt  when  it  amounted  to  $1341.60.  How  long  did  he  have 
the  use  of  the  money  ? 

15.  A  debt  of  $10,000  on  interest  at  5J%  amounted  to 
$10,618.75  when  it  was  paid.     How  long  had  it  run  ? 


INTEREST  177 

311.     Written 

1.  What  principal  on  interest  at  6  %  will  gain  |90  in  1  yr. 

1  mo.  10  da.  ? 

Statement  of  Relation  :  Principal  x  xf  fy  x  f  f^  =  1 90. 
Which  terms  of  relation  are  given  ?    Which  is  to  be  found  ? 

Solution 
1 

15 
Principal  =  90  --  (^f  ^  x  |§§)  =  90  -  ^V  =  #  1350  Ans. 

2.  What  principal  will  earn  $80  in  two  years  at  5%  ? 

3.  A  farmer  owed  a  debt  on  which  he  paid  1495  interest  in 
three  years,  the  rate  being  5^  % .     How  much  did  he  owe  ? 

4.  A  certain  city  borrowed  money  at  3|%  interest,  with 
which  to  build  a  city  hall.  In  7  yr.  6  mo.  the  city  paid 
178,750  interest  on  this  debt.  How  much  money  was  bor- 
rowed ? 

5.  What  principal  will  yield  $26.40  interest  in  1  yr.  4  mo. 

at  81%? 

6.  What   principal,  at  7%,  will  bring  1153.93  interest  in 

2  yr.  6  mo.  ? 

7.  A  man  paid  146.41  for  the  use  of  a  sum  of  money  for 
7  mo.  11  da.     The  rate  was  7  %.     What  was  the  principal? 

8.  A  man  paid  §209  interest  on  a  sum  of  money  for  9  mo. 
15  da.     If  the  rate  was  5i  % ,  what  was  the  principal  ? 


178  GRAMMAR  SCHOOL   ARITHMETIC 

312.    Written 

1.  What  principal  will  amount  to  1584.65  in  1  yr.  18  da. 
at6%? 

63 

$lx-^x|^  =  8.063  interest  on  f  1  for  1  yr.  18  da. 

100    ppP 
W 

10  $1,063  amount  of  |1  for  1  yr.  18  da. 

Statement  of  Relation:  $1,063  x  Principal  =  1584.65. 
Which  term  of  relation  is  to  be  found  ?    Find  it. 

2.  What  principal  will  amount  to  $431.20  in  2  yr.  at  6  %  ? 

3.  What  principal  on  interest  at  5%  will  amount  to  1 430  in 
1  yr.  6  mo.  ? 

4.  Mr.  Smith  borrowed  a  sum  of  money  at  4|  %  interest  for 
eight  months.  When  the  debt  became  due,  he  had  to  pay 
$2060.     What  was  the  sum  borrowed  ? 

5.  A  farmer  bought  a  hay  press,  agreeing  to  pay  for  it  in 
six  months,  with  5%  interest  on  the  purchase  price.  When 
the  money  became  due,  it  took  $491.20  to  settle  the  bill. 
What  was  the  purchase  price  ? 

6.  Mr.  Jacobs  bought  a  house  Nov.  23,  1905,  paying  three 
fifths  of  the  price  in  cash  and  the  remainder  with  5%  interest 
on  the  5th  of  February,  1907,  when  it  required  $1696  to  cancel 
the  debt. 

a.    How  much  was  left  unpaid  at  the  time  of  purchase  ? 
h.    What  was  the  purchase  price  of  the  house  ? 

7.  A  dealer  in  real  estate  offered  me  a  lot  for  $1317.50,  to  be 
paid  15  mo.  after  date  of  purchase,  without  interest.  This  was 
equal  to  what  cash  price,  money  being  worth  6  %  ? 


INTEREST 


179 


313.    Written 

In  examples   1-20  find  the  terms  indicated   hy  interrogation 
points : 


Principal 

Rate 

Time 

Interest 

Amount 

1. 

$364.24 

6% 

1  yr.  4  mo. 

? 

2. 

12700 

5% 

1  yr.  1  mo. 

? 

3. 

12350 

5% 

1  yr.  3  mo.  6  da. 

? 

? 

4. 

1292 

H% 

90  da. 

Exact 

? 

5. 

$1730 

4% 

? 

$318.32 

6. 

? 

4i% 

2  yr.  9  mo. 

$1556.775 

? 

7. 

$387.50 

? 

7  mo.  24  da. 

$20.15 

8. 

$3500 

5% 

? 

$630 

9. 

$1000 

e% 

? 

? 

$2000 

10. 

$250 

? 

90  da. 

? 

$252.50 

11. 

$3500 

? 

July  18  to  Nov.  9 

$70 

12. 

? 

5% 

Jan.  1  to  May  25, 

1908 

? 

$3580 

13. 

? 

5|% 

? 

$132 

$4132 

14. 

$1800 

? 

Feb.  20  to  Sept.  21, 
1907 

$86.75 

15. 

? 

6%' 

6  mo.  6  da. 

$494.88 

16. 

$620.50 

5i% 

30  da. 

Exact 

? 

17. 

? 

5% 

146  da. 

Exact 

$765 

18. 

$800 

? 

73  da. 

Exact 

$811.20 

19. 

$2500 

5% 

? 

Exact 

$2550 

20. 

$350 

51% 

Apr.  1  to  Nov.  6 

Exact 

? 

180  •  GRAMMAR   SCHOOL   ARITHMETIC 

21.  What  is  the  difference  between  the  exact  interest  and  the 
common  interest  on  $657  for  90  da.  at  5%  ? 

22.  How  long  will  it  take  11440  to  earn  1244.80  interest 
at4J%? 

23.  What  sum  must  a  lady  have  invested  at  5  %  per  annum 
to  yield  her  an  income  of  S125  a  month  ? 

COMPOUND   INTEREST 

314.  Compound  interest  is  interest  computed  hy  adding  the 
unpaid  interest  to  the  principal  at  regular  interest  periods^  and 
talcing  the  sum  for  a  new  principal  for  each  succeeding  interest 
period. 

315.  Simple  interest  is  interest  computed  on  the  original  prin- 
cipal for  the  entire  time. 

In  ordinary  business  transactions,  "  with  interest "  is  under- 
stood to  mean  simple  interest,  although  the  debt  may  run  for 
several  years. 

It  is  customary  to  insert  in  contracts  for  the  payment  of 
interest,  where  the  debt  runs  for  a  longer  period  than  one  year, 
a  provision  that  the  interest  shall  be  paid  at  regular  periods, 
usually  of  three  months,  six  months,  or  one  year.  This  is 
especially  true  in  the  case  of  insurance  companies,  loan  asso- 
ciations, and  other  institutions  doing  a  large  loan  business ;  so 
that  they  are  enabled  to  compute  their  income  on  a  compound 
interest  basis  by  loaning  the  interest  as  fast  as  it  is  paid  in. 

Savings  banks  and  trust  companies  generally  allow  compound 
interest  on  all  deposits  remaining  for  a  full  interest  period,  which 
is  usually  three  or  six  months. 

316.  Written 

1.    Find  the  compound  interest  of  1350  for  2  yr.  and  6  mo. 

at  6%, 


COMPOUND  INTEREST  181 

Solution 

$350.00  Principal 

21.00  Interest  for  1st  year 
$371.00  Amount  taken  as  new  principal 

22.26  Interest  for  2d  year 
$393.26  Amount  used  as  new  principal 

11.80  Interest  for  6  mo. 
$405.06  Amount  for  2  yr.  6  mo. 
350.00  1st  principal 
$55.06  Compound  interest  for  2  yr.  6  mo. 

Note  1.  —  When  the  interest  is  compounded  semi-annually,  the  rate  for 
each  period  is  one  half  the  annual  rate ;  when  quarterly,  one  fourth. 

When  no  interest  period  is  mentioned,  interest  is  compounded  annually. 

Note  2.  —  In  actual  practice,  compound  interest  is  computed  by  means 
of  compound  interest  tables  similar  to  that  on  page  410.  The  table  gives  the 
amounts  of  one  dollar  for  from  one  to  twenty  periods,  at  various  rates  for 
each  period.  The  required  amount  is  obtained  by  multiplying  the  amount 
of  one  dollar,  for  the  required  number  of  interest  periods,  at  the  given  rate, 
by  the  given  principal.  If  the  compound  interest  is  desired,  omit  the  1 
at  the  left  of  the  decimal  point  in  the  multiplicand. 

2.  What  is  the  compound  interest  of  $830  for  3  years  at  5  %  ? 

3.  What  is  the  amount  of  $  650  for  4  years  at  4  %  interest, 
compounded  semiannually  ? 

4.  What  is  the  compound  interest  of  1365  for  2  yr.  7  mo.  18 
da.  at  6  %,  compounded  semiannually  ? 

5.  What  is  the  compound  interest  on  1640  for  4  years  at 
5%? 

6.  What  is  the  interest,  compounded  quarterly,  on  f  538.25 
for  2  yr.  6  mo.,  rate  4  %  ? 

7.  What  is  the  interest,  compounded  annually,  on  $683.48 
for  4  years  at  6  %  ? 

8.  What  is  the  compound  interest  on  1437.50,  for  3  yr.  6 
mo.,  at  5  %,  compounded  semiannually? 


182 


GRAMMAR   SCHOOL   ARITHMETIC 


PROMISSORY  NOTES 

317.  A  promissory  note  is  a  written  promise  made  hy  one  party 
to  pay  absolutely  a  specified  sum  of  money  to  another  party  at  a 
certain  time. 

Since  the  term  "note"  in  business  transactions  always  refers 
to  a  promissory  note,  we  shall  henceforth  omit  the  word  "promis- 
sory" in  speaking  of  a  note. 

FORMS  OF  NOTES 

318.  The  following  forms  illustrate  various  kinds  of  notes : 

Note  I 


^600^^  Springfield,  Mass.,  CCucf.  d,  1^07 

.. _^^_^?^_^_f^ _^^after  date,  J  promise  to  pay 

to  trie  order  of Z 7_ 

.Z^:^?/'f!f!!f^.^^.'?^rrr:::rrrrrrr:r::r.A)ZZars,  with  interest. 

Value  received. 


Note  I.— Back 


^ 

1 

^ 

^ 

^ 

^ 

^ 

■■^^ 

^ 

^^ 

^ 

^ 

^ 

^ 

PROMISSORY   NOTES 
Note  2 


183 


Los  Angeles,  Cat.,  (Z^vvt  /,  /(^08 


U'yb&  u&ci'b 


1ft 


after  date,  J  projnise  to  pay. 


W^.    f.    fSoAA. 


■3^MH>  ku.rvcLv&ci 


or  Nearer :?:rrrrrr?:?:^:_^^/:r::___  nYf_rrrr:::r:::doiia.rs, 

with  interest  at  seven  per  cent. 
Value  received. 


Note  2.  —  Back 


^         ^ 

^         ^ 

^  ^ 

%.     -> 

s   ^ 

V     ^%, 

^  ^ 

^^    ^ 

^   ^ 

oj  -3 

^  ^ 

^  ^ 

^ 

-^-^ 

v:  ^^ 

d  * 

^  i" 
^  J 
^  S 

Note  3 


f/000^^ 

dn  d&yyuam^ci 


Scranton,  Pa.,  Tflciif  12,  f^ifOS 
prormse  to  pay L-i- 


for  value  received,  with  interest. 


184  GRAMMAR   SCHOOL   ARITHMETIC 

Note  3.  —  Back 


Note  4 


to  the  order  of'. 


Cleveland,  O.,  c/fo-v-.  f ,  /(^07 
after  date,  J  promise  to  pay 

.dollars.     Value  received. 


Note  4.  — Back 


PROMISSORY  NOTES  185 

Note  5 


/^i'JU^  Rochester,  K.Y.,  incuy.  /,  /(J08 

?^^^__^^_^_^ after  date,  J  promise 

to  pay / 

Value  received. 


KINDS   OF   NOTES 

319.  The  party  who  makes  the  promise  is  the  maker  of  a  note. 

320.  The  party  to  whom  the  money  is  promised  to  he  paid  is 
the  payee  of  a  note. 

321.  The  party  who  owns  a  note  is  the  holder. 

322.  The   sum  promised  to  he  paid.,  not  including  interest.,  is 
the  face  of  a  note. 

323.  A  note  in  which  the  maker  promises  to  pay  interest  is  an 
interest-bearing  note. 

324.  A  note  in  which  the  maker  does  not  promise  to  pay  in- 
terest is  a  no n- interest- bearing  note. 

325.  A  time  note  is  a  note  pay  ahle  at  a  specified  time  after  date. 

326.  A  demand  note  is  a  note  pay  ahle  on  demand  of  the  holder. 
A  note  payable  one  day  from  date  becomes  a  demand  note, 

for  the  holder  may  require  payment  at  any  time  after  date. 

327.  A  note  is  negotiable  (i.e.   transferahle)  when  it  is  drawn 
pay  ahle  — 

a.    To  the  hearer^     h.    To  the  payee  or  hearer^  or     c.    To  the 
order  of  the  payee. 


186  GRAMMAR  SCHOOL   ARITHMETIC 

Note.  —  Besides  the  previous  conditions,  a  note  to  be  negotiable — 
In  Alabama,  must  be  payable  at  a  fixed  place. 
In  Indiana,  must  be  payable  at  a  bank. 
In  West  Virginia,  must  be  payable  at  a  banking  office. 

328.  A  note  is  non-negotiable  when  it  is  drawn  payable  only  to 
the  payee. 

329.  A  note  should  contain  the  following  things,  in  addition 
to  the  words  of  the  promise: 

a.  The  time  and  place  at  which  the  note  is  made. 

h.  The  face,  expressed  both  in  figures  and  in  words. 

c.  The  name  of  the  payee. 

d.  The  time  of  payment. 

e.  The  name  of  the  maker. 

/.    The  words  "  with  interest,"  and  the  rate,  if  the  note  is 
intended  to  be  interest  bearing. 
g,    "  Value  received." 

A  note  is  valid  without  the  words  "  Value  received,"  but 
there  is  a  legal  advantage  in  using  them. 

Note.  —  There  are  many  kinds  of  notes,  such  as  "  joint "  notes,  "  joint  and 
several"  notes,  "judgment"  notes,  "collateral"  notes,  and  others,  which 
are  not  in  general  use  and  involve  legal  distinctions  that  do  not  come  within 
the  scope  of  elementary  arithmetic.  Hence  no  treatment  of  them  is  here 
given. 

INDORSEMENT 

330.  An  indorsement  is  a  name  or  other  writing  on  the  bach  of 
a  note.     Usually  an  indorsement  contains  either 

a.  The  name  of  the  payee,  or  of  some  other  person  or  persons, 
or, 

b.  A  record  of  payments  made  on  the  note. 

331.  A  person  indorses  a  note  in  blank  by  merely  writing 
his  name  across  the  back  of  it. 


PROMISSORY  NOTES  187 

332.  A  person  indorses  a  note  in  full  by  writing  "  Pay  to  the 

order  of "  (the  name  of  the  person  to  whom  the  note 

is  transferred)  and  signing  his  name  below. 

333.  One  who  indorses  a  note  is  called  an  indorser. 

334.  One  to  whose  order  a  note  is  made  payable  hy  the  indorse- 
ment is  called  the  indorsee. 

335.  An  indorser,  by  the  act  of  indorsement,  agrees  to  pay 
the  note  when  due  if  the  maker  does  not ;  but  an  indorser  may 
avoid  this  liability  for  the  payment  of  a  note  by  writing 
"  Without  recourse  "  above  his  signature. 

336.  When  the  payee  of  a  note  drawn  payable  to  the  payee's 
order  transfers  the  note,  he  must  indorse  it  in  order  to  make  it 
payable  to  the  new  holder. 

If  he  indorses  it  in  hlanh^  it  becomes  payable  to  the  holder, 
whoever  he  may  be,  and  can  be  transferred  again  without 
further  indorsement.  If  he  indorses  it  in  full,  it  becomes  pay- 
able only  to  the  person  designated  in  the  indorsement,  until  it 
is  in  turn  indorsed  by  that  person. 

He  may  make  a  restrictive  indorsement,  by  writing  over  his 

signature  "  Pay  to "  (naming  some  person).     With 

such  an  indorsement,  the  note  cannot  again  be  transferred,  for 
it  is  payable  only  to  the  person  designated.  A  restrictive 
indorsement  is  sometimes  written,  "  Pay  to oi^ly-" 

MATURITY 

337.  The  day  on  which  a  note  becomes  due,  or  payable,  is  the 
day  of  maturity. 

In  most  states,  a  note  becomes  due,  or  payable,  on  the  day 
specified  for  payment  in  the  note ;  in  a  few  states,  the  note 
does  not  become  due  until  three  days  after  the  time  specified 
in  the  note.     These  three  da3^s  are  called  day&  of  grace.     The 


188  GRAMMAR  SCHOOL  ARITHMETIC 

debtor's  legal  right  to  days  of  grace  has  been  recognized  by 
the  courts  because  of  the  prevalent  custom,  in  early  times,  of 
allowing  this  extra  time  for  payment. 

The  present  tendency  is  toward  a  restriction  of  the  custom, 
and  the  states  are,  one  by  one,  enacting  laws  abolishing  days  of 
grace. 

If  a  note  falls  due  on  Sunday,  or  a  legal  holiday,  it  is  gener- 
ally not  collectible  until  the  next  business  day.  In  a  few  states 
it  becomes  due  on  the  last  preceding  business  day.  In  New 
York  State,  a  note  falling  due  on  Saturday  is  not  collectible 
until  the  following  Monday. 

If  no  time  of  payment  is  mentioned  in  a  note,  it  is  payable 
on  demand. 

DEFAULT   OF  PAYMENT 

338.  When  the  maker  of  a  note  fails  to  pay  it  on  the  day  of 
maturity,  it  is  the  duty  of  the  holder  to  notify  the  indorsers  of 
that  fact.  If  they  are  not  so  notified  within  a  reasonable  time, 
they  are  freed  from  liability  for  its  payment.  Can  you  think 
of  some  reason  for  this  rule  ? 

When  the  maker  does  not  pay  a  note  on  the  day  of  maturity, 
the  indorser  may  pay  it  and  then  collect  it  from  the  maker. 
When  there  are  several  indorsers,  and  the  maker  fails  to  pay 
the  note  when  due,  the  first  indorser  may  pay  it  and  sue  the 
maker ;  or  any  other  indorser  may  pay  it  and  sue  the  maker 
and  all  the  previous  indorsers. 

EXERCISES 

339.  Oral 

1.  From  the  forms  on  pages  182-185  select,  giving  reasons 
for  the  selections, — 

a.  A  time  note. 

b.  A  demand  .note. 


PROMISSORY  NOTES  189 

c.  A  negotiable  note. 

d.  A  non-negotiable  note. 

e.  A  note  that  may  be  transferred  without  indorsement. 
/.  A  note  that  cannot  be  transferred  without  indorsement. 
g.  A  note  that  cannot  be  transferred. 

h.    An  indorsement  in  blank. 

i.    An  indorsement  in  full. 

j.  An  indorsement  that  does  not  make  the  indorser  liable 
for  payment  of  the  note. 

h.  An  indorsement  that  makes  the  note  transferable  again 
without  further  indorsement. 

1.  A  note  that  is  partly  paid. 
m.    An  interest-bearing  note. 

n.  A  non-interest-bearing  note. 

2.  Name  the  maker  of  each  note. 

3.  Name  the  payee  of  each  note. 

4.  Name  the  indorser  of  each  note. 

5.  Who  can  collect  note  1  ? 

6.  If  C.  F.  Harper  sells  note  1,  what  must  he  do  to  make  it 
payable  to  the  one  who  buys  it  ? 

7.  Who  can  collect  note  3  ? 

8.  Who  can  collect  note  5  ? 

9.  Who  is  liable  for  the  payment  of  note  1  ? 
10.    Who  is  liable  for  the  payment  of  note  4  ? 

340.     Written 

Pupils  number  around  the  class,  "  one,  two,  three ;  one,  two. 
three,"  etc,  until  each  pupil  has  a  number. 

1.  a.  Each  of  the  number  I's  write  a  note  that  can  be  trans- 
ferred only  by  being  indorsed,  making  himself  the  maker,  and 
number  2  the  payee. 


190  GRAMMAR  SCHOOL  ARITHMETIC 

h.    Deliver  the  note  to  number  2. 

e.    Number  2  transfer  the  note  to  number  3,  indorsing  it  in  full. 

d.  Number  3  transfer  the  note  to  the  teacher,  indorsing  it  so 
that  the  teacher  may  transfer  it  again  without  indorsing  it. 

e.  To  whom  may  the  teacher  look  for  payment  of  the  note  ? 
/.   Number  1  is  which  party  ?    Number  2  ?    Number  3  ?    The 

teacher  ? 

2.  a.  Each  of  the  number  3's  write  a  note  payable  to  num- 
ber 2  or  bearer. 

h.    Deliver  the  note  to  number  2. 

c.  Number  2  transfer  the  note  to  number  1. 

d.  Number  1  transfer  the  note  to  the  teacher. 

e.  How  many  indorsements  are  necessary  in  making  these 
transfers  ? 

/.    To  whom  may  the  teacher  look  for  payment  ? 

g.  Both  number  1  and  number  2  might  have  indorsed  the 
note.  Would  their  indorsement  in  blank  have  affected  the 
value  of  the  note  ?     If  so,  how  and  why  ? 

3.  a.  Number  2  write  a  note  payable  to  number  1  or  order. 
h.    Deliver  it  to  number  1. 

c.  Number  1  transfer  it  to  number  3,  indorsing  it  in  full. 

d.  Number  3  transfer  it  to  the  teacher,  indorsing  it  without 
recourse. 

e.  To  whom  can  the  teacher  look  for  payment  ? 

4.  a.  Every  pupil  write  a  non-negotiable  demand  note  bear- 
ing interest  at  the  legal  rate  where  made,  making  the  teacher 
the  payee. 

h.    Deliver  the  note. 

c.  Who  can  collect  the  note  ? 

d.  Who  must  pay  the  note  ? 

e.  How  could  a  third  party  become  liable  for  the  payment 
of  the  note  ? 


COMPUTING  INTEREST   ON   NOTES  191 

COMPUTING  INTEREST  ON  NOTES 

341.  An  interest-bearing  note  bears  interest  from  the  day  of 
date  to  the  day  of  payment. 

A  non-interest-bearing  note,  if  not  paid  at  maturity^  bears 
interest  from  the  day  of  maturity  until  paid,  at  the  legal  rate 
where  made. 

If  no  rate  of  interest  is  mentioned  in  an  interest-bearing  note, 
interest  must  be  computed  at  the  legal  rate  in  the  state  in  which 
the  note  is  made. 

342.  The  face  of  a  note  is  the  principal. 

343.  The  sum  of  the  principal  and  interest  is  the  amount  of 
the  note. 

344.  When  the  time  mentioned  in  a  note  is  expressed  in 
months,  calendar  months  are  always  understood.  Thus,  a  note 
for  three  months  given  July  15  is  due  Oct.  15,  or,  where  grace 
is  allowed,  Oct.  18.  A  90-day  note  given  July  15  is  due  90 
days  after  July  15,  or  Oct.  13. 

345.  Written 

1.  Find  the  amount  of  note  2,  page  183. 

2.  Find  the  amount  of  note  1,  page  182,  the  legal  rate  of 
interest  in  Massachusetts  being  6  % . 

3.  Find  the  amount  of  note  3,  page  183,  if  paid  on  the 
third  day  of  January,  1909,  the  legal  rate  of  interest  in  Penn- 
sylvania being  6  % . 

4.  Find  the  amount  of  note  4,  page  184,  if  not  paid  until 
Aug.  11,  1908,  the  legal  rate  of  interest  in  Ohio  being   6%. 

5.  How  much  can  Mr.  Walden  collect  on  note  5,  page  185, 
if  it  is  paid  Aug.  20,  1908,  the  legal  rate  of  interest  in  New 
York  being  6  %  ? 


192  GRAMMAR   SCHOOL  ARITHMETIC 

6.  A  demand  note  for  1711  with  interest  was  dated  at 
Ogden,  Utah,  July  7,  1905,  and  paid  Sept.  30,  1905.  How 
much  was  paid,  the  legal  rate  of  interest  in  Utah  being  8%  ? 

7.  A  90-day  note  for  $960,  with  interest  at  7%,  was  made 
July  1,  1906,  at  Lincoln,  Neb.,  where  grace  is  allowed. 

a.    On  what  day  did  the  note  mature  ? 
h.    How  much  was  due  at  maturity  ? 

8.  A  60-day  note  for  $1200  without  interest,  dated  at 
Cairo,  111.,  Jan.  1,  1904,  was  not  paid  until  May  15,  1904. 
What  sum  was  then  due,  the  legal  rate  of  interest  in  Illinois 
being  5  %  ? 

9.  Find  the  amount  at  maturity  of  a  30-day  interest-bear- 
ing note  for  $700  in  the  state  where  you  live. 

10.  What  must  be  the  face  of  a  90-day  note  that  will  amount 
to  $263.90,  computing  interest  at  6%,  without  grace? 

11.  Find  the  amount  at  maturity  of  the  following  note,  the 
rate  of  interest  in  Louisiana  being  5  %  and  grace  being  allowed  : 

1 600 J>oV-  New  Orleans,  Sept.  1,  1908. 

On  the  15th  day  of  December,  1908,  I  promise  to  pay  to  the 
order  of  Henry  P.  Emerson,  six  hundred  dollars,  with  interest. 
Value  received.  John  H.  Gardner. 

12.  Write  a  note  for  $1000  that  will  give  James  Thorne  the 
right  to  collect  $1020  from  you  90  days  from  the  date  of 
the  note. 

13.  Find  the  amount  due  June  15  on  an  unpaid  non-interest- 
bearing  30-day  note  for  $  250,  dated  March  3,  in  a  state  where 
the  legal  rate  of  interest  is  6  %. 

14.  Write  a  negotiable  note  dated  at  your  city  or  town,  Jan. 
15,  due  May  7  of  the  present  year,  and  find  the  amount  duo 
at  maturity. 


PARTIAL  PAYMENTS  193 

i 

PARTIAL   PAYMENTS 

346.  When  payments  are  made  in  sums  less  than  the  entire 
amount  of  a  note,  the  holder  indorses  them  on  the  back  of  the 
note,  and  they  are  known  as  indorsements,  or  partial  payments. 

The  rule  given  below  is  the  one  adopted  by  the  Supreme 
Court  of  the  United  States  for  determining  the  amount  due  on 
a  debt  on  which  partial  payments  have  been  made.  It  is  the 
legal  rule  in  most  of  the  states  of  the  Union.  Classes  in  any 
state  having  a  different  rule  should  follow  the  legal  rule  of 
their  own  state,  in  solving  the  partial  payment  problems  given 
in  this  book. 

United  States  Rule  for  Partial  Payments 

347.  Find  the  amount  of  the  debt  to  the  time  when  a  payment, 
or  the  sum  of  the  payments,  equals  or  exceeds  the  interest  due,  and 

from  that  amount  subtract  such  payment  or  sum  of  payments. 
With  this  remainder  for  a  7iew  principal,  proceed  as  before  to  the 
time  of  settlement. 

This  rule  means  that  neither  the  whole  interest  nor  any  part 

of  it  shall  be  used  to  increase  the  principal  on  which  interest  is 

paid;   but  whenever  more  than  enough  to  cover  the   interest 

has  been  paid,  the  excess  shall  be  used  to  diminish  the  principal. 

348. 


//^i'^ —  Watertown,  JV.T.,  fa^yv.    /,   /(^06 

Hn  cUyyvc^yici,    for   value   received,    J  promise  to  pay 
to   the   order   of, jZ.    A    S^cLVoxym. 

w~iZk  iyyiZ&v&at.  RaiyeA^t   of.     lA}kit& 


194 


GRAMMAR  SCHOOL   ARITHMETIC 


-I 


■Of 

1 


b-      ^       ^ 
^       ^      ^ 


V:> 

K 

^ 

^ 

vv 

^V 

V 

\ 

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lo 

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^ 

^\ 

^ 

^ 

r 

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^ 


The  diagram  at  the 
left  shows  a  part  of 
the  back  of  the  note,  on 
page  193,  on  which  in- 
dorsements were  made 
as  given.  The  amount 
due  at  date  of  settle- 
ment is  computed  be- 
low by  the  United 
States  rule. 


Note.  —  The  legal  rate  of  interest  in  New  York  State  is  6  per  cent. 

Subtracting  each  date  from  the  one  above  to  find  interest 
periods : 

Remainders 

L  yr.  ^tJi  Int.  per, 

3  mo.  6  da.,  or  96  da.,  ^th  Bit,  per. 
2  mo.  12  da.,  or  72  da.,  Sd  Int.  per. 
8  mo.  0  da.,  or  240  da.,  2c?  Int.  per. 

4  mo.  24  da.,  or  144  da.,  1st  Int.  per. 
12  =  2  yr.  6  mo.  12  da.,    Proof  of  int.  periods 

Subtracting  /  from  a,  we  obtain  2  yr.  6  mo.  12  da.,  which  is 
the  same  as  the  sum  of  the  remainders.  This  proves  that  the 
interest  periods  are  correct. 

24 

$1820       ^       I^^_ 

^l~''ioo'' 


Yr. 

Mo. 

Da. 

a. 

1908 

7 

13 

h. 

1907 

7 

13 

c. 

1907 

4 

7 

d. 

1907 

1 

25 

e. 

1906 

5 

25 

/. 

1906 

1 

1 

2 

6 

12 

$43.68 

1820.00 

$1863.68 

250.00 
$1613.68 


Interest  for  1st  period 
First  principal 
Amount 
First  payment 
New  principal 


PARTIAL,  PAYMENTS 


195 


4 
11613.68  ^_$__^m^ 
1  100      ^^ 

12 


$64.55     Interest  for    2d  period  ex- 
ceeds payment 


S1613.68  ..    0    ^  7;2 


^  100  ^  ^^p 


10 
16 


19.36 

I169T.59 

420.00 

11277.59 


$1277.59  .,    ^ 


^ibo"^ 


$548.03 


349.    Written 


10 

^ioo" 


Interest  for  Sd  period 

Amount 

Sum  of  2d  and  ?>d  payments 

New  principal 

=        20.44  Interest  for  ith  period 

$1298.03  Amount 

750.00  4th  payment 

$  548. 03  JVew  principal 

32.88  Interest  for  5th  period 

$580.91  Due  at  date  of  settlement 

Ans. 


1.  Write  a  demand  note  for  $792  with  interest,  dated  Jan.  15, 
1902,  at  Springfield,  111.     Indorse  payments  as  follows:  Dec.  15, 

1902,  $50;  Aug.  30,  1903,  $12.50;  Oct.  25,  1903,  $155.  Find 
the  amount  due  Dec.  1,  1903,  computing  interest  at  5%. 

2.  A  note  without  interest,  dated  Lexington,  Ky.,  Aug.  15, 

1903,  promising  to  pay  $1200  thirty  days  from  date,  has  $200 
indorsed  Nov.  16,  1903,  and  $350,  March  4,  1904.  How  much 
was  due  April  1,  1904,  the  legal  rate  in  Kentucky  being  6  %  ? 

3.  What  was  due  March  1, 1901,  on  a  note  for  $  1000  with  inter- 
est at  9  %,  dated  March  1,  1900,  with  indorsements  as  follows : 
Aug.  10,  1900,  $300;  Sept.  1,  1900,  $100;  Jan.  1, 1901,  $50? 

4.  What  amount  was  necessary  to  settle,  Oct.  20, 1905,  a  note 
for  $2000,  with  interest  at  6%,  dated  July  20,  1903,  bearing 
indorsements  of  $700,  Sept.  10,  1903,  and  $75,  Oct.  20,  1904? 


196  GRAMMAR   SCHOOL   ARITHMETIC 

5.  A  note  for  $700  with  interest  at  7  %  was  given  Dec.  12, 
1906.  Payments  of  $200,  Dec.  12,  1907,  and  $159,  April  5, 
1908,  were  made.     What  was  due  Oct.  30,  1908  ? 

6.  How  much  was  due  Aug.  1, 1906,  on  a  note  for  -1380,  with 
interest  at  5%,  dated  Aug.  1,  1904,  on  which  were  indorsed 
payments  of  1 15,  May  30,  1905,  and  $90,  Jan.  1,  1906? 

7. 

$300  Troy,  N.  Y. ,  Oct.  12,  1899 

On  demand,   for   value   received,  I  promise    to   pay 

^^.^.^.^.^.^.^.^.^-^.^^S.  D.  Cleveland or  order,   Three  hundred 

dollars,  with  interest. 

J.  H.  Van  AUtyne. 

The  following  payments  were  made  on  this  note :  June  27, 
1901,  1150;  Dec.  9,  1902,  $150.     What  was  due  Oct.  9,  1905? 

8.  On  a  note  for  $573.25,  with  interest  at  6  %,  dated  June  10, 
1900,  were  the  following  indorsements :  April  5,  1901,  $14.30; 
July  14,  1902,  $250.     How  much  was  due  Sept.  20,  1903  ? 

9.  A  note  of  $850  was  dated  June  21, 1902,  bearing  interest 
at  6%.  On  this  note  were  the  following  indorsements:  Sept. 
15, 1902,  $150.90  ;  Nov.  21, 1903,  $45;  Jan.  15, 1904,  $256.88. 
What  remained  due  June  21,  1904  ? 

10.  A  man  bought  a  farm,  Jan.  1,  1901,  giving  in  part  pay- 
ment a  bond  and  mortgage  for  $1900,  due  on  demand,  with 
interest  at  ^%.  He  paid  $40,  July  1,  1901;  $300,  Feb.  15, 
1902;  and  $240,  July  20,  1902.  How  much  was  due  at  time 
of  settlement,  Jan.  1,  1903  ? 

11.  On  a  note  for  $832.26  dated  Aug.  3,  1899,  the  following 
payments  were  indorsed:  $350,  Oct.  5,1900;  $468.37,  May 
15,  1902.     How  much  was  due  Dec.  12,  1903,  interest  at  7  %  ? 


PARTIAL   PAYMENTS  197 

12.  Face,  $2950.  Date,  July  1,  1905.  Interest,  7%.  In- 
dorsements; Oct.  1,  1905,  1750;  Jan.  15,  1906,  1600;  Dec.  1, 
1906, 1300;  March  1, 1907,  $450.     What  was  due  July  1, 1907? 

350.  When  notes  and  accounts,  upon  which  partial  payments 
have  been  made,  are  settled  within  a  year  after  interest  begins, 
business  men  sometimes  make  use  of  the  following 

Merchants'  Rule 

Find  the  amount  of  the  entire  debt  at  date  of  settlement. 
Find  the  amount  of  each  payment  at  date  of  settlement. 
Subtract  the  amount  of  the  payments  from  the  amount  of  the  debt. 

351.  Written 

Find  the  balance  due  at  time  of  settlement  on  each  of  the  debts  in 
examples  1-5,  using  the  Merchants'  Rule. 

1.  A  note  for  $700,  dated  Jan.  1,  1904.  Indorsements: 
$215,  April  15;  $124.68,  April  30 ;  $21.04,  July  7;  $130,  Oct. 
20.     Settled  Jan.  1,  1905.     Rate  5  %. 

2.  A  note  for  $250,  dated  March  31,  1906.  Indorsements: 
$10.45,  July  1,  1906;  $130,  Dec.  4,  1906;  $50,  Jan.  1,  1907. 
Settled  Feb.  28,  1907.     Rate  6%. 

3.  A  debt  of  $1240  contracted  July  1,  1907,  with  payments 
of  $280,  Jan.  1, 1908,  and  $135,  April  15, 1908.  Settled  June  1, 
1908.     Rate4|%. 

4.  $700  borrowed  Oct.  22,  1905;  payments  made  Jan.  1, 
1906,  and  March  14, 1906,  of  $280.50  and  $35.90  respectively. 
Settled  April  1,  1906.     Rate  7  %. 

5.  A  man  bought  a  house  for  $5500,  May  1,  1904,  paying 
$4000  at  that  time  and  $200  the  15th  of  each  month,  besides 
interest  at  5  %.     Settled  Dec.  1,  1904. 


J98  GRAMMAR   SCHOOL   ARITHMETIC 

REVIEW  AND  PRACTICE 
352.      Oral 

1.  ReadMCMXII;  305.0070100;      \^9    . 

2.  Name  three  powers  of  10  ;  two  powers  of  5. 

3.  Count  by  12's  to  132. 

4.  What  terra  in  division  corresponds  to  the  product  in 
multiplication  ? 

5.  State  two  ways  of  testing  subtraction. 

6.  State  two  ways  of  testing  division. 

7.  How  many  decimal  places  does  the  quotient  contain  ? 

8.  Give  results  rapidly,  adding  tens  first : 

28  +  35  ;  46  +  43  ;  53  + 17  ;  82  +  49. 

9.  2x15-14-2  +  3x10  =  ? 

10.  What  problems  can  be  solved  by  cancellation? 

11.  How  can  we  tell,  without  actually  dividing,  whether  a 
number  is  divisible  by  3  ?  By  9  ?  By  2  ?  By  4  ?  By  8  ?  By  5  ? 
By  25?    By  6? 

12.  Give  results  at  sight :  247-^100;  .83546  x  1000;  36  x 
25;  497.60-^100;  9 -.- 25. 

13.  Without  actually  dividing,  tell  whether  247,658  will  ex- 
actly divide  42,130,071,359,  and  why. 

14.  Name  two  composite  numbers  that  are  prime  to  each 
other, 

15.  How  may  we  tell  whether  a  number  is  prime  or  not  ? 

16.  How  may  we  find  a  single  divisor  that  will  reduce  a 
fraction  to  lowest  terms  ? 

17.  When  is  a  number  in  its  simplest  form  ? 

18.  What  fractions  cannot  be  reduced  to  exact  decimals  ? 


REVIEW  AND  PRACTICE  199 

19.  Using  aliquot  parts,  answer  the  following  questions  : 

a.  At  25  cents  a  pound,  what  will  5Q  pounds  of  coffee  cost  ? 

b.  How  many  packages  of  cereal,  at  12J  cents  per  package, 
will  110  buy? 

c.  If  the  average  price  of  the  melons  in  a  load  is  16f  cents 
apiece,  what  are  48  of  them  worth  ?  How  many  can  be  bought 
for  $5? 

d.  At  14|  cents  a  dozen,  how  many  dozen  pencils  will  $2 
buy  ?     What  will  28  dozen  cost  ? 

20.  How  many  pence  are  there  in  20  shillings  ? 

21.  Alice  bought  half  a  ream  of  note  paper.  How  many 
sheets  of  paper  did  she  buy  ?     How  many  quires  ? 

22.  Name  four  kinds  of  figures  that  are  quadrilaterals. 

23.  How  many  cords  are  there  in  a  pile  of  2-foot  wood,  12 
ft.  long  and  8  ft.  high  ? 

24.  Describe  a  board  foot  and  with  your  hands  show  its  size. 

25.  How  many  feet  of  lumber  are  there  in  a  piece  of  scant- 
ling 8' by  4'' by  3''? 

26.  How  many  cubic  inches  of  oil  are  there  in  3  gal.  ? 

27.  What  per  cent  is  equal  to  i  ?  |?  |?  -jQ^  ?  f?  -f?  f?  |? 

28.  18%  of  1200=?     80%  of  20?     66|%ofl2oz.? 

29.  $17  is  25%  of  what? 

30.  What  is  170  %  of  10  quarts  ? 

31.  What  per  cent  of  $65  is  f  13  ? 

32.  What  is  J  %  of  800  miles  ? 

33.  1^  %  of  49  gallons  are  how  many  gallons  ? 

34.  ^  of  49  gallons  are  how  many  gallons  ? 

35.  1  sq.  ft.  is  what  per  cent  of  1  sq.  yd.  ? 

36.  Three  quarts  are  what  per  cent  of  one  gallon  ? 


200  GRAMMAR  SCHOOL   ARITHMETIC 

37.  80  %  of  £1  =  how  many  shillings  ? 

38.  A  man  earns  $40  a  week  and  spends  60%  of  it.  How 
much  does  he  save  ? 

39.  Frank  missed  three  problems  in  a  lesson  of  15  problems. 
What  per  cent  of  the  lesson  did  he  have  correct  ? 

40.  Mr.  Peck  sold  a  piano  bench  for  <|13,  gaining  $3.  What 
per  cent  did  he  gain  ? 

41.  A  furniture  dealer  bought  a  chair  for  f  20  and  gained 
40  %  on  it.     What  was  the  selling  price  ? 

42.  Upon  what  base  are  gain  and  loss  always  computed  ? 

43.  A  merchant  sold  a  bill  of  goods  for  §40  more  than  they 
cost,  thereby  gaining  20  % .     What  did  the  goods  cost  ? 

44.  A  merchant  sold  a  bill  of  goods  for  $20  more  than  they 
cost,  thereby  gaining  10%.     What  was  the  selling  price? 

45.  A  merchant  sold  a  bill  of  goods  for  $24,  thereby  gaining 
20  %.     What  did  the  goods  cost  ? 

46.  A  man  sold  a  horse  for  $180,  thereby  losing  10%. 
What  did  he  pay  for  the  horse  ? 

47.  What  per  cent  is  gained  on  chestnuts  bought  at  $1.20 
per  peck  and  sold  for  20  cents  a  quart  ? 

48.  What  per  cent  is  lost  on  chestnuts  bought  at  20  cents  a 
quart  and  sold  at  $1.20  a  peck  ? 

49.  A  man  sold  $1500  worth  of  goods  on  a  commission  of 
10%.     How  much  should  he  pay  over  to  his  principal  ? 

50.  An  agent's  commission  at  12  %  for  selling  a  consignment 
of  goods  amounted  to  $48.  What  was  the  value  of  the  goods 
sold  ?     How  much  did  the  principal  receive  from  the  sales  ? 

51.  A  real  estate  agent's  commission  at  2  %  for  selling  a 
business  block  was  $800.  What  was  the  selling  price  of  the 
block  ? 


REVIEW  AND   PRACTICE  201 

52.  The  net  amount  of  a  bill  which  had  been  discounted  5  % 
was  $9.50.  What  was  the  face  of  the  bill?  What  was  the 
discount? 

53.  What  is  the  net  amount  of  a  bill  of  1200  on  which  com- 
mercial disi;ounts  of  20%  and  10%  have  been  made? 

54.  Whai>  single  discount  is  the  same  as  two  successive  dis- 
counts of  Ip  %  each  ? 

55.  Wha'  is  the  premium  for  insuring  a  house  for  $2000 
for  three  ye^ars  at  the  rate  of  70  cents  per  $100  of 
insurance  ?      , 

56.  How  miiv^h  is  saved  on  $1000  of  insurance  for  three  years 
by  taking  a  thr.  e-year  policy  at  1  %  instead  of  three  one-year 
policies  at  J  %  (  ich  ? 

57.  When  a  ai^rglar  insurance  policy  for  $1000  costs  $12.50 
per  year,  the  /  ^mium  is  what  per  cent  of  the  face  of  the 
policy  ? 

58.  Of  what  three  factors  is  interest  the  product  ? 

59.  Define  a  promissory  note. 

60.  Describe  a  negotiable  note. 

61.  Describe  a  non-negotiable  note. 

62.  How  may  a  note  be  indorsed  in  blank  ? 

63.  By  indorsing  a  note  in  blank,  what  contract  does  the 
indorser  make  ? 

64.  How  may  a  person  indorse  a  note  so  as  to  avoid  liability 
for  its  payment  ? 

65.  Give  the  United  States  rule  for  partial  payments. 

66.  Give  the  Merchants'  Rule  for  partial  payments. 


202 


GRAMMAR  SCHOOL  ARITHMETIC 


353.    Written 

The  following  table,  compiled  from  the  records  of  -he  United 
States  Weather  Bureau,  shows  in  degrees  the  average  tempera- 
ture for  each  month  in  twenty  different  places: 


Jan. 

P"eb. 

Mak. 

Apr. 

May 

June 

July 

Aug. 

Sept, 

jOCT. 

60 

Nov. 
46 

Dec. 

Albany,  N.Y. 

31 

32 

40 

56 

69 

78 

82 

80 

72 

36 

Atlanta,  Ga. 

50 

54 

61 

70 

79 

85 

87 

85 

81 

71 

60 

53 

Baltimore,  Md. 

41 

43 

49 

61 

73 

82 

86 

84 

77 

66 

53 

44 

Binghamton,  N.Y. 

33 

30 

41 

55 

67 

77 

83 

80 

7': 

63 

45 

35 

Bismarck,  N.  Dak. 

17 

20 

32 

54 

67 

75 

82 

81 

cu 

56 

37 

25 

Boston,  Mass. 

35 

36 

42 

54 

66 

76 

81 

78 

J' 

60 

49 

39 

Carson  City,  Nev. 

44 

48 

53 

61 

67 

76 

84 

84  .   T5 

65 

56 

46 

Cincinnati,  0. 

40 

43 

51 

63 

74 

83 

87 

84    ■  78 

m 

52 

43 

Galveston,  Tex. 

59 

62 

68 

74 

81 

86 

89 

88   ,j84 

78 

68 

62 

Harrisburg,  Pa. 

36 

36 

45 

60 

71 

80 

84 

82  , .  75 
84^1^77 

62 

49 

40 

Indianapolis,  Ind. 

36 

39 

48 

62 

72 

82 

86 

64 

49 

40 

Jacksonville,  Fla. 

64 

67 

72 

78 

84 

89 

91 

9^^.  86 

78 

71 

65 

Minneapolis,  Minn. 

24 

24 

36 

58 

69 

78 

83 

'i.^ 

74 

60 

39 

28 

New  Orleans,  La. 

61 

65 

70 

76 

82 

87 

89 

88 

85 

78 

69 

63 

Portland,  Me. 

30 

32 

39 

51 

62 

72 

77 

75 

68 

57 

45 

35 

Portland,  Ore. 

44 

48 

55 

61 

67 

71 

78 

77 

71 

62 

52 

47 

St.  Louis,  Mo. 

40 

43 

52 

66 

75 

84 

88 

86 

79 

68 

53 

44 

Santa  Fe,  N.M. 

39 

43 

52 

60 

69 

78 

81 

79 

73 

62 

50 

43 

Spokane,  Wash. 

33 

38 

48 

59 

68 

74 

83 

83 

71 

59 

44 

37 

Yuma,  Ariz. 

66 

72 

78 

85 

93 

101 

106 

104 

100 

87 

76 

68 

1-12.    Find  the  average  temperature  of  all  the  places  for  each 
month. 

13-32.    Find  to  the  nearest  hundredth  of  a  degree  the  average 
annual  temperature  of  each  place. 

33.  Express  in  Roman  numerals  the  number  of  the  present 
year. 

34.  Multiply  in  the  shortest  way  : 

a.  39,742,568  by  25.        h.  34,067  by  125.        c.  394,708  by  99. 


REVIEW  AND  PRACTICE  203 

35.  Divide  in  the  shortest  way : 

a,  39,763  by  25.         h.  9834  by  125.         c.  796,453  by  .16f. 

36.  Resolve  7511  into  its  prime  factors. 

37.  ReducB  111^  to  lowest  terms. 

38.  Find  the  smallest  number  that  will  exactly  contain  39, 
36,  and  84. 

39.  Find  tne  greatest  number  that  will  exactly  divide  2205 
and  3024. 

40.  Reduce  Iff  to  a  decimal. 

41.  Write  a  bill  containing  two  debit  items  and  one  credit 
item.     Foot  the  bill  and  receipt  it  as  clerk  for  the  creditor. 

42.  A  field  i  mile  long  and  30  rods  wide  contains  how  many 
acres  of  land  ? 

43.  15s.  9d.  are  what  part  of  one  pound  ? 

44.  Divide  18°  17'  30''  by  15. 

45.  Multiply  7  hr.  40  min.  8  sec.  by  15. 

46.  What  is  the  cost,  at  28  cents  a  square  yard,  of  painting 
the  walls  and  ceiling  of  a  room  33  ft.  by  24  ft.  and  11  ft.  high, 
allowing  for  five  windows,  each  4  ft.  by  8  ft.,  and  three  doors, 
each  4  ft.  by  8  ft.  6  in.  ? 

47.  Find  the  cost  of  the  following  bill  of  lumber : 

30  scantlings,       18'  x  2''  x    4"  at  126  per  M. 

40  joists,  16'  X  3"  X  12"  at  126  per  M. 

25  joists,  16'  X  2"  X  10"  at  1 26  per  M. 

120  boards,  14'  x  |"  x    4"  at  1 35  per  M. 

300  pieces  siding,  10'  x  |^"  x    5"  at  $55  per  M. 

48.  A  box  car  36  ft.  long  and  8J  ft.  high  contains  102  cu. 
yd.  of  space.     How  wide  is  it  ? 

49.  12.80  is  ^  per  cent  of  what  sum  ? 


204  GRAMMAR  SCHOOL  ARITHMETIC 

50.  I  of  50  bu.  are  how  many  quarts  ? 

51.  Three  days  are  what  per  cent  of  two  weeks  ? 

52.  A  speculator  bought  150  crates  of  eggs  in  April  and  May, 
paying  15  j^  a  dozen  for  one  third  of  them  and  17)^  a  dozen  for 
the  remainder.  Each  crate  contained  30  dozen  eggs.  In 
December,  he  sold  them  at  a  uniform  price  of  25 i^  a  dozen, 
and  out  of  the  profits  paid  a  bill  of  45/  per  crate  for  cold 
storage,  and  $13.80  for  cartage  and  other  expenses.  What  per 
cent  net  profit  did  he  make  ? 

53.  During  the  month  of  December,  at  a  certain  place,  there 
were  8  stormy  days  and  22  cloudy  days,  the  remaining  days 
being  fair. 

a.    What  per  cent  of  the  days  were  stormy  ? 
h.    What  per  cent  were  cloudy  ? 
c.    What  per  cent  were  fair  ? 

54.  A  bill  of  goods  listed  at  |700  was  sold  at  a  discount  of 
15%,  12%,  and  5%.     Find  the  net  price. 

55.  a.  Find,  by  the  United  States  rule,  the  balance  due  at 
settlement  on  a  debt  of  1630,  contracted  April  1,  1907,  and 
settled  Sept.  1,  1908,  on  which  payments  of  $15.50,  Dec.  11, 
1907,  and  1125.00,  Feb.  16,  1908,  had  been  made.  Interest 
allowed  at  5%. 

h.  If  this  balance  were  computed  by  the  Merchants'  Rule, 
would  it  favor  the  debtor  or  the  creditor,  and  how  much  ? 

56.  A  bill  of  hardware  was  discounted  80%,  10%,  and  5%, 
and  then  amounted  to  13.42.     What  was  the  list  price  ? 

57.  Which  is  the  better  offer,  successive  discounts  of  30%, 
10  %,  and  5  %,  or  successive  discounts  of  5  %,  10  %,  and  30%  ? 

58.  Which  is  the  better  offer,  successive  discounts  of  15%, 
5  %,  and  2  %,  or  successive  discounts  of  20  %  and  2  %  ? 


BANKS   AND  BANKING  205 

59.  The  premium  for  one  kind  of  accident  insurance  policy  is 
at  the  rate  of  1 5  per  ilOOO.  The  agent's  commission  is  30% 
of  the  premium. 

a.  What  is  the  face  of  the  policy  for  which  the  company 
receives  f  17.50  after  paying  the  agent's  commission? 

h.    What  is  the  face  of  a  policy  that  yields  the  agent  f  3.75  ? 
c.    What  is  the  agent's  commission  on  a  16000  policy  ? 

60.  A  bill  of  $20  was  reduced  by  three  successive  discounts. 
If  the  first  two  discounts  were  20  %  and  10  %,  and  the  net  price 
was  $12.96,  what  per  cent  was  the  third  discount? 

BANKS  AND   BANKING 

354.  There  are  many  kinds  of  banking  institutions,  but  most 
of  them  may  be  included  in  three  general  divisions ;  viz.  savings 
banks,  banks  of  deposit,  and  trust  companies. 

355.  Savings  banks  are  designed  to  be  safe  places  of  deposit 
for  small  sums  of  money.  These  sums  are  usually  the  savings 
of  people  who  have  not  the  inclination  or  opportunity  to  en- 
gage in  large  business  enterprises.  Savings  banks  pay  a  low 
rate  of  interest  on  all  balances  of  one  dollar  or  more,  and  the 
interest  is  compounded  quarterly,  semi-annually,  or  annually. 
The  interest  is  computed  by  means  of  tables,  and  each  bank  has 
its  own  method  of  calculation. 

In  order  that  the  money  of  depositors  may  be  safeguarded, 
savings  banks  are  generally  forbidden  by  law  to  make  loans 
unless  secured  by  mortgages  on  real  estate,  and  from  making 
investments,  except  in  special  kinds  of  property,  such  as  gov- 
ernment bonds  and  bonds  of  certain  states  and  cities. 

356.  Banks  of  deposit,  otherwise  known  as  commercial  banks, 
or  banks  of  discount,  transact  a  much  wider  range  of  business 


206  GRAMMAR  SCHOOL  ARITHMETIC 

than  do  savings  banks.  They  may  loan  money  on  notes,  collect 
accounts  and  notes  for  customers,  issue  bills  of  exchange  and 
letters  of  credit,  and  make  many  kinds  of  investments  which 
savings  banks  are  not  permitted  to  make.  As  a  rule  they  pay 
no  interest  on  deposits,  but  the  services  that  they  render  to 
their  customers  are  considered  sufficient  compensation  for  their 
use  of  the  money  on  deposit. 

Banks  of  deposit  which  are  organized  under  Federal  laws 
and  are  under  the  supervision  of  the  United  States  govern- 
ment are  known  as  national  banks ;  those  that  are  organized 
according  to  state  laws  and  are  under  the  supervision  of  state 
authorities  are  generally  known  as  state  banks,  though  each 
individual  bank  adopts  a  name  of  its  own. 

State  and  national  banks  transact  in  general  the  same  kinds 
of  business ;  but  national  banks  also  perform  a  special  function 
in  connection  with  the  issuance  of  paper  money,  which  will  be 
considered  later. 

357.  Trust  companies  are  similar  in  some  respects  to  savings 
banks,  and  in  other. respects  to  banks  of  deposit. 

They  resemble  savings  banks  in  that  they  pay  interest  on 
deposits.  They  are  generally  not  allowed  to  loan  money  on 
notes,  except  when  secured  by  collateral,  i.e.  some  specific 
piece  of  property,  put  into  the  hands  of  the  trust  company  to 
be  sold  by  the  company  if  the  note  is  not  paid  when  due. 

Otherwise  they  are  much  like  banks  of  deposit,  having  in 
some  respects  even  greater  latitude  in  the  kinds  of  business 
which  they  may  transact. 

DEPOSITING   AND  WITHDRAWING  MONEY 

368.  One  who  has  money  on  deposit  in  a  hank  is  called  a 
depositor. 


BANKS   AND  BANKING 


207 


DEPOSIT  SLIP 


When  a  person  deposits  money  for  the  first  time  in  any  par- 
ticular bank,  he  receives  from  the  bank  a  book  in  which  he  is 
credited  with  the  sum  deposited. 

A  depositor  in  a  savings  bank  takes  his  book  with  him  when- 
ever he  deposits  or  withdraws  money.  To  deposit  money  he 
merely  hands  it  to  the  receiving  teller,  who  credits  in  the  bank 
book  the  amount  of  the  deposit.  To  withdraw  money,  he 
hands  his  book  to  the  pay- 
ing teller,  and  signs  a  re- 
ceipt for  the  money  to  be 
withdrawn.  The  teller 
charges  in  the  bank  book 
the  amount  withdrawn  and 
paj'^s  it  to  the  depositor. 

In  depositing  money  in 
any  other  bank  than  a  sav- 
ings bank,  the  depositor  fills 
out  a  deposit  slip  stating  in 
separate  items  the  amount 
of  paper  money,  of  gold,  of 
silver,  and  of  checks  which 
he  deposits.  This  slip  is 
handed  in  with  the  money 
and  checks  deposited,  and  is  used  by  the  teller  in  making  up 
his  balance  at  the  close  of  the  day's  business. 

Withdrawals  from  a  bank  of  deposit  are  made  by  means  of 
checks. 

A  check  is  a  written  order^  signed  hy  a  depositor^  directing  the 
hank  to  pay  to  a  certain  person^  or  to  his  order^  or  to  the  hearer^ 
a  specified  sum  of  money. 

When  the  bank  pays  the  sum  directed  to  be  paid,  it  charges 
the  depositor's  account  with  the  amount  paid. 


MARINE  NATIONAL  BANK 

OF  BUPFALO 

Deposited  to  Credit  of" 

Buffalo,  N.Y.           Cl^a^.^3      IOCS' 

qiiRRFuny 

DOLLARS 

CTS. 

/as 

3S^ 
I  ^3 

7S 
70 

anin, 

Sll  VFR^ 

CHFO.KH, 

/xXi  ^  ^y^  /S^^J^, 

Ga^a^aJH^. 

AMf^l'iVT, 

3  6S 

8-5 

208  GRAMMAR  SCHOOL   ARITHMETIC 


Dace     yi^^-^  I  ISO  S 

rORDER  OF 


BUFFALO.N.Y 


OF    BUFFALO. 


^Xpul.  o^"*-^/^ 


~   Dollars 


4f^A.a£cLi^(^^^tih 


stub  Check 

359.  The  amount  named  in  a  check  is  called  the  face. 

360.  The  depositor  who  signs  a  check  is  called  the  drawer  of 
the  check. 

361.  The  person  to  whom^  or  to  whose  order,  a  check  is  made 
payable  is  called  the  payee. 

362.  The  hank  on  which  a  check  is  drawn  is  called  the  drawee. 
In  the  above  check  which  party  is  John  White  ?     Gerald  W. 

Porter  ? 

Every  depositor  in  a  bank  of  deposit  receives  from  the  bank 
a  check  book,  which  consists  of  blank  checks  bound  together, 
each  check  attached  to  a  stub  as  shown  above.  When  a  check 
is  filled  out,  the  stub  is  filled  out  to  agree  with  it,  and  the  check 
is  then  torn  off,  through  the  perforated  line.  When  all  the 
checks  have  been  used,  there  remains  a  book  of  stubs  contain- 
ing a  record  of  all  the  checks,  the  number  of  each  check,  its 
date,  its  face,  the  name  of  the  payee,  and  the  purpose  for  which 
it  was  used.  Some  check  books  are  so  arranged  that  the  stub 
may  also  show  the  balance  remaining  in  the  bank  after  each 
check  is  drawn. 

Checks  are  convenient  in  paying  bills;  for  by  means  of 
them  the  depositor  may  avoid  carrying  or  sending  money.  To 
illustrate,  let  us  suppose  that  Mr.  A,  a  merchant  in  Cleveland, 


BANKS  AND  BANKING  209 

buys  a  bill  of  goods  from  Mr.  B,  in  Chicago.  A  fills  out  a 
check  payable  to  B's  order  and  mails  it  to  B.  B  indorses  the 
check,  deposits  it  in  his  own  bank  at  Chicago,  and  it  is  credited 
on  his  account.  The  banks  attend  to  the  rest  of  the  business. 
The  check  is  finally  returned  to  A's  bank  in  Cleveland,  and 
the  amount  is  charged  to  A's  account,  and  credited  to  the 
account  of  B's  bank  in  Chicago. 

Most  banks  make  a  practice  of  returning  all  checks  to  deposi- 
tors. These  checks,  being  indorsed  in  each  case  by  the  payee, 
serve  as  receipts  for  the  amounts  paid. 

363.  If  the  drawer  of  a  check  is  a  stranger  to  the  payee,  the 
payee  may  be  unwilling  to  accept  the  check  in  lieu  of  money, 
fearing  that  the  maker  may  not  have  money  on  deposit  sufficient 
to  pay  the  check  when  presented  at  the  bank  for  payment. 
Then  the  maker  may  be  required  to  have  the  check  certified. 
To  do  this,  he  takes  the  check  to  the  bank,  and  the  bookkeeper, 
teller,  or  other  proper  person  stamps  on  its  face  the  word  "  cer- 
tified "  with  the  name  of  the  bank,  and  writes  in  his  own  name. 
He  then  makes  a  memorandum  of  the  amount  on  the  drawer's 
account.  The  bank  is  then  obliged  to  cash  the  check  when 
presented.  The  certification  of  the  check  is  equivalent  to  the 
bank's  promise  to  pay. 

A  COMPARISON  OF  CHECKS  AND  NOTES 

364.  1.  A  note  is  a  promise  to  pay  money,  while  a  check  is 
an  order  to  pay  money. 

2.  A  check  always  has  three  parties,  while  a  note  may  have 
only  two. 

3.  A  check,  like  a  note,  may  be  negotiable  or  non-negotiable, 
according  to  the  manner  in  which  it  is  drawn. 


210  GRAMMAR  SCHOOL   ARITHMETIC 

4.  A  negotiable  check  may  be  transferred  hy  indorsement  in 
the  same  manner  as  a  note,  and  the  indorser  is  liable  for  its 
payment  if  it  is  not  paid  by  the  maker  or  drawee. 

5.  The  different  forms  of  indorsement  have  the  same  force 
when  made  on  a  check  as  when  made  on  a  note. 

6.  A  note  may  draw  interest,  but  a  check  does  not. 

EXERCISES 

365.  Oral 

1.  Name  some  similarities  or  differences  between  a  check  and 
a  note,  other  than  those  given  above. 

2.  Who  is  the  drawer  of  the  check  on  page  208  ? 

3.  Who  is  the  payee  ?     The  drawee  ? 

4.  Tell  whether  the  check  is  negotiable  or  non-negotiable. 

5.  How  must  a  check  be  worded  in  order  to  be  negotiable  ? 

366.  Written 

Let  the  pupils  of  the  class  take  numbers  one,  two,  and  three, 
as  on  page  189.     Let  the  teacher  be  the  First  National  Bank. 

1.  Number  two  make  out  a  bill  against  number  one^  and  re- 
ceipt it  when  paid. 

2.  Number  one  write  a  negotiable  check  and  give  it  to  num- 
ber two  in  payment  of  the  bill. 

3.  Number  Uvo  indorse  the  check  to  number  three  and  take  a 
receipt  for  the  amount  on  account. 

4.  Number  three  indorse  the  check  in  blank  and  deposit  it  to 
his  own  credit  in  the  bank. 

5.  Teacher  mark  "  paid  "  and  return  to  number  one. 

Note.  —  Repeat  this  and  similar  exercises  until  pupils  are  familiar  with 
the  use  of  checks. 


BANK  DISCOUNT  211 

BANK  DISCOUNT 
367.    A  note  that  is  payable  to  or  at  a  hank  is  a  bank  note. 


$  ^°^  ^  Syracuse.N.Y.       ^^  \o         iQQ^ 

*  W'(^XXuO'-i~K^     . ■ ~        AFTER  HATE    •/    PROXISE 

TO  \»Xi  TO  THE  ORDER  OF  MMAaX-V    5cUAjL^Al\ju.*t^_-^ 

SpkJUojj^  — — '^^ 

National  Bank  of  Syracuse  ,)    ^//^Kixl^ a.,  }w^JUa^^ 

SYRACUSE,Ny.  ^  " 


DOU^ARS 

too 


AT 

V^LUE  Received 


368.    Banks  come  into  possession  of  notes  in  two  ways  : 
a.    They  may  lend  money  directly  to  the  maker  and  take  his 

note,  or, 

h.    The  note  may  be  drawn 

"S       ^       ^        .  i    payable   to   another  party  and 

"^     ^      ^      ^  \   be  bought  by  the  bank,  or  de- 

S    ^       ^      ^  \     P<^sited  in  the  bank  for  collec- 


tion 
^  ^  \^         Either  of  these  ways  is  equiv- 

<^    -^e  ;:^  y     alent  to  a  purchase  of  the  note 

by  the  bank.    When  a  bank  thus 


buys  a  note,  it  pays  less  than  the  maturity  value;  hence  the 
transaction  is  called  discounting  the  note. 

369.  The  sum  deducted  from  the  maturity  value  of  a  note  in 
determining  the  price  to  he  paid  for  the  note  hy  a  hayik  is  called  the 
bank  discount. 

370.  The  sum  paid  for  a  note  hy  a  hank,  or  the  difference  he- 
tween  the  maturity  value  and  the  hank  discount,  is  called  the  pro- 
ceeds of  the  note. 


212  GRAMMAR  SCHOOL   ARITHMETIC 

371.  The  day  on  which  a  note  is  discounted  is  called  the  day 
of  discount. 

372.  The  time  from  the  day  of  discount  to  the  day  of  maturity 
is  the  term  of  discount. 

373.  If  the  bank  should  buy  the  note  in  §  367  on  the  day  of 
date,  the  proceeds  would  be  determined  as  follows : 

Day  of  maturity,  Jan.  8,  1908. 

Day  of  discount,  Oct.  10,  1907. 

Term  of  discount,  90  days. 

Interest  on  $  500  for  90  days  at  6%,  I  7.50. 

$  500  -  1 7.50  =  $ 492.50.     Proceeds, 

If  the  bank  should  buy  the  note  Nov.  19,  the  proceeds  would 
be  determined  as  follows  : 

Day  of  maturity,  Jan,  8,  1908. 

Day  of  discount,  Nov.  19,  1907. 

Term  of  discount  (Nov.  19,  1907,  to  Jan.  8,  1908), 

50  days. 
Interest  on  $500  for  50  days  at  6%,  14.17. 
$500-14.17  =  1495.83.     Proceeds. 

In  determining  the  proceeds  of  an  interest-bearing  note,  the 
general  practice  of  banks  is  to  find  the  amount  of  the  note 
at  maturity  and  compute  the  interest  on  that  amount  for  the 
term  of  discount.  That  interest  is  the  bank  discount.  The 
bank  discount  subtracted  from  the  maturity  value  (which  is  the 
amount  in  this  case)  gives  the  proceeds. 

374.  In  all  cases  we  may  apply  the  following : 

Rule  for  finding  the  bank  discount  and  proceeds  of  a  bank 
note. 

1.  Find  the  amount  due  at  maturity.  This  is  the  maturity 
value. 


BANK  DISCOUNT  213 

2.  Find  the  time  from  the  day  of  discount  to  the  day  of  matu- 
rity.     This  is  the  term  of  discount. 

3.  Find  the  interest  on  the  maturity  value  for  the  term  of 
discount.     This  is  the  hank  discount. 

4.  Subtract  the  hank  discount  from  the  maturity  value  to  find 
the  proceeds. 

Note  1. — When  the  time  mentioned  in  a  note  is  given  in  months,  calen- 
dar months  are  understood.  For  example,  a  note  dated  July  12,  payable 
three  months  after  date,  is  due  Oct.  12,  or  92  days  after  date. 

Note  2. — In  most  states,  notes  falling  due  on  Sunday  or  a  legal  holiday 
are  payable  on  the  next  business  day,  and  interest  and  discount  are 
reckoned  to  that  day. 

Note  3. — In  states  allowing  days  of  gi*ace,  the  date  of  maturity  is  three 
days  later  than  the  time  mentioned  in  the  note,  and  the  term  of  discount 
three  days  longer  than  when  grace  is  not  allowed. 

The  local  practice  in  regard  to  holidays,  days  of  grace,  etc.,  should  be 
followed  in  solving  problems. 

375.    Oral 

1.  How  is  the  maturity  value  of  an  interest-bearing  note 
found  ? 

2.  How  does  the  maturity  value  of  an  interest-bearing  note 
compare  with  the  face  of  the  note  ? 

3.  How  does  the  maturity  value  of  a  non-interest-bearing 
note  compare  with  the  face  of  the  note,  if  paid  when  due  ? 

4.  A  30-day  note  is  dated  Jan.  15.  What  is  the  day  of  ma- 
turity ? 

5.  A  60-day  note  was  dated  Feb.  20,  1908.  When  did  it 
mature  ? 

6.  Mr.  Field,  wishing  to  borrow  from  a  bank,  made  out  a 
60-day  bank  note  for  1 100  without  interest,  dated  Sept.  11, 
1907.     What  was  the  date  of  maturity  ?     How  much  was  due 


214  GRAMMAR   SCHOOL   ARITHMETIC 

at  maturity  ?  If  Mr.  Field  had  his  note  discounted  on  the  day 
of  date,  what  was  the  term  of  discount  ?  What  was  the  dis- 
count, the  legal  rate  being  6  %  ? 

7.  Mr.  Brown  bought  a  horse  from  Mr.  Martin,  giving  in 
payment  a  bank  note  for  $200  without  interest,  dated  July  9, 

1906,  payable  90  days  from  date.  On  the  8th  day  of  August, 
Mr.  Martin  indorsed  the  note  and  deposited  it  in  the  bank,  re- 
ceiving credit  for  the  proceeds.  What  was  the  day  of  ma- 
turity ?  The  day  of  discount  ?  The  term  of  discount  ?  The 
bank  discount,  the  legal  rate  being  69^?  How  much  was 
credited  to  Mr.  Martin's  account  ? 

8.  A  bank  note  for  f  500,  without  interest,  due  in  90  days, 
dated  May  7,  1905,  was  discounted  June  6,  1905.  What  were 
the  proceeds,  money  being  worth  6  %  ? 

9.  A  note  for  $400,  bearing  interest  at  7  %,  dated  Jan.  1, 

1907,  and  due  in  90  days,  was  discounted  on  the  day  of  date. 
What  was  the  maturity  value  ?  On  what  sum  was  the  dis- 
count computed  ? 

376.     Written 

1.  A  man  gave  his  note  for  $  720  for  90  days  without  in- 
terest. What  was  it  worth  at  a  bank  where  the  discount  rate 
was  6  %? 

2.  How  much  can  I  borrow  from  a  bank  by  giving  my  60- 
day  note  for  $650  without  interest,  if  the  bank  gives  me  a  dis- 
count rate  of  5  %  ? 

3.  A  merchant  bought  a  piano  for  $400  cash  and  sold  it  the 
same  day,  taking  in  payment  a  90-day  bank  note  for  $500, 
which  he  immediately  indorsed  and  deposited  in  his  bank,  re- 
ceiving credit  for  the  proceeds  at  a  discount  rate  of  7%  per 
annum.     What  was  his  profit  on  the  piano? 


BANK  DISCOUNT  215 

4.  What  were  the  proceeds  of  a  note  for  $300  without  in- 
terest, due  Jan.  7,  1907,  and  discounted  Nov.  15,  1906,  the  dis- 
count rate  being  5  %  ? 

5.  The  following  note  was  discounted  at  the  rate  of  4J  % 
per  annum  on  the  21st  day  of  January,  1905.  What  were  the 
proceeds  ? 

19600  New  York,  December  7,  1904. 

Ninety  days  after  date  I  promise  to  pay  to  the  order  of  the 
New  York  National  Exchange  Bank  nine  thousand  six  hundred 
dollars. 

Value  received.  Chakles  H.  Redmond. 

6.  What  are  the  proceeds  of  a  six-months  note  for  $800, 
without  interest,  dated  May  7,  1903,  and  discounted  Oct.  15, 
1903,  at  the  rate  of  6  %  per  annum  ? 

7.  A  man  in  Seattle  accepted  a  30-day  note  for  $  975,  without 
interest,  in  payment  for  furniture.  Nine  days  later  he  had  the 
note  discounted  at  the  rate  of  8  %  per  annum.  What  did  he 
receive  for  it  ? 

8.  Silas  Brown  sold  a  vacant  lot  on  the  15th  day  of  April, 
1906,  to  James  Otis,  taking  in  part  payment  a  six-months  note 
for  1900  without  interest,  signed  by  Francis  Fernald,  dated 
Dec.  1,  1905,  and  payable  to  Mr.  Otis  at  the  Marine  Bank. 
Mr.  Otis  indorsed  the  note  to  Mr.  Brown's  order  and  Mr. 
Brown  immediately  indorsed  it  in  blank  and  had  it  discounted. 
The  discount  rate  was  7  % . 

a.    Write  the  note  and  make  all  the  indorsements. 
h.    How  much  did  Mr.  Brown  receive  for  the  note  ? 

9.  A  90-day  note  for  $1000  with  interest  at  6  %  was  dis- 
counted at  6  %  on  the  day  of  date.    What  were  the  proceeds  ? 

10.    On  the  first  day  of  March,  1907,  Edward  F.  Jones  bor- 
rowed  $800  from   John   Ethridge,   giving   his   note   for   one 


216  GRAMMAR   SCHOOL   ARITHMETIC 

year  with  interest  at  8%,  payable  at  the  Corn  Exchange 
Bank.  On  the  first  day  of  January,  1908,  Mr.  Ethridge  had 
the  note  discounted  at  6%  per  annum.  How  much  did  he 
receive  for  it? 

11.  A  90-day  note  for  1690,  bearing  interest  at  6%,  was  dis- 
counted at  the  same  rate  60  days  after  date.  What  were 
the  proceeds  ? 

12.  A  merchant  sold  at  25  %  profit  a  bill  of  goods  that  cost 
him  $150  cash,  taking  in  payment  a  60-day  note  without  interest, 
which  he  had  discounted  immediately  at  7  %  per  annum.  What 
was  his  net  profit  on  the  bill  of  goods? 

13.  A  farmer  received  $297  as  the  proceeds  of  a  note,  with- 
out interest,  due  in  60  days,  discounted  at  6  %  per  annum. 
What  was  the  face  of  the  note  ? 

Solution 

Discount  for  60  da.  =  1  %  of  face.  (4 «  x  f-i^  =  -01,  or  1%.) 

Proceeds  for  60  da.  —  99  %  of  face. 
Statement  of  Relation :  99  %  of  face  =  $  297. 

14.  I  borrowed  1591  from  a  bank,  giving  my  note  for  90  da. 
without  interest,  the  rate  of  discount  being  6  % .  What  was 
the  face  of  the  note  ? 

15.  Edward  H.  Flint  gave  William  G.  Barrows  his  note, 
without  interest,  payable  30  days  after  date  at  the  Third 
National  Bank.  Mr.  Barrows  indorsed  the  note  and  deposited 
it  on  his  account  on  the  day  of  date,  receiving  credit  for 
8477.20,  the  rate  of  discount  being  1  %  .  Write  the  note  and 
indorse  it  properly. 

16.  Robert  M.  Smith  borrowed  1715.26  from  the  Security 
National  Bank,  giving  his  note  for  100  days,  without  interest, 


BANK   DISCOUNT  217 

which  was  discounted  at  7%,  and  indorsed  by  Fred  Howard. 
Write  the  note  and  indorse  it. 

17.  A  farmer  gave  in  payment  for  farm  machinery  a  bank 
note  for  $600,  due  six  months  from  date,  without  interest, 
money  being  worth  8%.  That  was  equivalent  to  how  much  in 
cash? 

18.  Mr.  Walsh  owed  $700  at  the  bank.  When  it  became 
due,  he  obtained  30  days'  extension  of  time  by  paying  the  bank 
discount  for  that  time  at  the  rate  of  7  % .  How  much  did  he 
pay  to  secure  the  extension  ? 

19.  By  paying  $3.50,  a  debtor  obtained  a  15  days' extension 
of  time  on  a  debt  at  a  bank,  which  made  a  discount  rate  of  6  % . 
How  much  did  he  owe  ? 

Statement  of  Relation  ;  Face  x  xf  ij-  X  -^V^  =  $3.50. 

20.  What  are  the  proceeds  of  a  six-months  note  for  §400, 
bearing  interest  at  5%,  discounted  four  months  after  date 
at  6%? 

21.  What  is  the  face  of  a  non-interest-bearing  note  payable 
90  days  after  date  which  will  bring  $550  if  discounted  70  days 
after  date  at  6  %  ? 

22.  A  non-interest-bearing  note,  dated  May  7,  1904,  due  in 
three  months,  was  discounted  at  6%,  June  8,  1904,  yielding 
1574.20.     What  was  its  face? 

23.  Given  the  amount  $896.50,  term  of  discount  45  days, 
rate  of  discount  5|  %.     Find  the  proceeds. 

24.  Given  the  proceeds  $1541.99,  rate  of  discount  7  %,  time 
33  days.     Find  the  face. 

25.  Write  a  60-day  bank  note  without  interest,  which  will 
yield  enough,  if  discounted  at  6  %  on  the  day  of  date,  to  buy 
25  acres  of  land  at  $29.70  per  acre. 


218  GRAMMAR  SCHOOL   ARITHMETIC 

377.  PROTESTING  NOTES,  CHECKS,  AND  DRAFTS 

SYRACUSE,  N.Y.    ^OUn.     8,   1908 
SIR: 

PLEASE  TO   TAKE   NOTICE  that  a    nx)t&    made  by  <^lryuotki^    L. 

/ifuak&Q^  DATED  (^eZ.  /O,  1907,  FOR  ^600  AND  INDORSED  BY  YOU,  WAS  THIS 
DAY  PROTESTED  for  non-payment,  and  that  the  holders  look  TO  YOU  FOR 
THE    payment     THEREOF,    PAYMENT    HAVING     BEEN     DEMANDED    AND    REFUSED. 

YOURS    RESPECTFULLY, 

F.    L.    BARNES, 

NOTARY    PUBLIC. 

TO   ^kojvt&a.  ^l{y^ 

If  a  bank  note,  check,  or  draft  (see  page  231)  is  not  paid  at 
the  time  specified,  a  notice  similar  to  the  above  is  sent  to  each 
of  the  indorsers.  This  is  called  a  notice  of  protest,  and  sending 
it  is  called  protesting  the  note,  check,  or  draft. 

If  notice  of  protest  is  not  sent  within  a  reasonable  time  after 
default  in  payment  has  been  made,  the  indorsers  are  released 
from  liability  for  payment.  Banks  usually  protest  a  note  after 
banking  hours  on  the  day  of  maturity.  This  notice  enables  an 
indorser  to  protect  himself  and  avoid  needless  expense.  It  is 
customary  to  send  a  notice  of  protest  to  the  maker,  also,  though 
he  cannot  avoid  liability  for  payment  if  the  notice  is  not  sent. 

The  notice  of  protest  is  always  signed  by  a  notary  public,  who 
is  generally  an  officer  or  employee  of  the  bank,  also. 

Consult  your  dictionary  to  find  the  meaning  of  notary  public.  Most 
notaries  public  are  not  connected  with  banks. 

378.  Oral 

1.  Can  you  define  a  notice  of  protest  ? 

2.  Why  is  a  note  protested,  when  unpaid  at  the  time  of 
maturity  ? 


BANK  DISCOUNT  219 

3.  The  notice  given  above  is  the  one  that  would  have  been 
sent  to  Charles  Gibbs,  if  the  note  on  page  211  had  not  been  paid 
when  due.  To  what  other  persons  would  the  notice  have  been 
sent? 

4.  What  is  a  notary  public  ? 

5.  Name  the  men  who  are  responsible  for  the  payment  of  the 
note  mentioned  above,  if  it  is  properly  protested  when  due  and 
unpaid  ? 

6.  Who  is  responsible  for  its  payment,  if  not  protested  when 
due  and  unpaid  ? 

7.  Who  is  always  liable  for  the  payment  of  a  note  ? 

379.     Written 

1.  A  bank  note  for  $450,  dated  April  1,  1903,  payable  60 
days  after  date,  without  interest,  was  properly  protested  when 
due,  and  was  finally  paid  by  one  of  the  indorsers  on  the  29th  of 
August,  1903.  The  indorser  was  obliged  to  pay  a  fee  of 
il.25  for  protesting  the  note,  together  with  interest  at  7  %  on 
the  note  from  the  day  of  maturity.     How  much  did  he  pay  ? 

2.  If  the  note  on  page  211  was  paid  by  the  maker  Jan.  18, 
1908,  including  $1.25  for  protesting,  how  much  did  he  pay,  the 
legal  rate  of  interest  in  New  York  State  being  6  %  ? 

3.  A  bank  note  for  $1000,  without  interest,  became  due  and 
was  protested.  Six  days  later,  the  maker  took  up  the  note  by 
giving  a  new  note  for  the  same  sum  for  30  days,  with  a  new 
indorser,  and  paying  the  bank  discount  on  the  new  note  at  6%, 
interest  on  the  old  note  from  the  day  of  maturity  at  6%,  and 
the  charge  for  protesting,  which  was  $1.75.  How  much  did 
he  pay? 

4.  The  maker  of  a  bank  note,  without  interest,  paid  the  note 
30  days  after  maturity,  with  interest  at  6  %  from  the  day  of 


220  GRAMMAR  SCHOOL  ARITHMETIC 

maturity,  and  a  charge  of  f  1.50  for  protesting.    If  he  paid 
$604.50,  what  was  the  face  of  the  note? 

TAXES 

380.  The  support  of  a  town,  village,  city,  county,  state,  or 
national  government  requires  a  large  sum  of  money.  This 
money  is  used  for  many  purposes,  such  as  carrying  on  the 
schools,  keeping  roads  and  streets  in  good  condition,  paying 
the  salaries  of  public  officers,  constructing  bridges  and  public 
buildings,  and  taking  care  of  the  poor  and  unfortunate  who  are 
unable  to  care  for  themselves.  This  money  is  used  for  the 
benefit  of  all  the  people  and  the  protection  of  their  lives  and 
property.  Hence  all  the  people  are  required  to  contribute 
toward  paying  the  expense,  according  to  the  value  of  their 
property. 

In  some  places  each  male  citizen  over  twenty-one  years  of 
age  is  required  to  pay  a  certain  sum  toward  the  expenses  of  his 
town,  regardless  of  the  value  of  his  property. 

Can  you  think  of  some  expenses,  other  than  those  given 
above,  that  occur  in  your  city,  village,  or  town  for  which  all  the 
people  must  pay?  Can  you  tell  how  the  valuation  of  the 
property  belonging  to  any  person  is  determined  ?  Name  as 
many  different  kinds  of  property  as  you  can. 

381.  A  tax  is  a  sum  of  money  levied  upon  persons  or  property 
for  puhlic  use. 

382.  A  tax  levied  on  persons  is  a  poll  tax. 

383.  A  tax  levied  on  property  is  a  property  tax. 

384.  Personal  property  is  property  that  is  movable,  as  money, 
notes,  furniture,  books,  and  tools. 

385.  Real  property  is  immovable  property,  as  houses  and  lands. 


TAXES  221 

386.  Assessors  are  officers  chosen  to  make  a  list  of  the  taxable 
property/  of  a  city^  village^  or  town^  estimate  its  value,  and  appor- 
tion the  tax. 

387.  A  tax  budget  is  a  list  of  all  the  items  of  expense  in  carry- 
ing on  a  state,  county,  city,  or  other  g over 7iment  for  a  certain  time, 
usually  one  year,  or  in  carrying  on  a  department  of  such  govern- 
ment. From  this  is  deducted  the  income  (from  licenses,  fines, 
sale  of  privileges,  etc.)  and  the  poll  tax,  if  any,  to  find  the 
net  amount  of  the  budget. 

388.  An  assessment  roll  is  a  list  of  all  the  taxable  property  in 
a  town,  village,  or  city,  with  the  assessed  value  of  each  piece  of 
property. 

389.  The  tax  rate  is  the  decimal  which  shows  what  part  of  the 
assessed  valuation  is  required  for  taxes.  It  is  determined  by 
dividing  the  net  amount  of  the  tax  budget  by  the  entire  assessed 
valuation  of  all  the  property  upon  which   the  tax  is  levied. 

The  rate  is  generally  expressed  in  a  decimal  of  four,  five,  or 
six  places,  showing  the  part  of  a  dollar  taken  as  the  tax  on  one 
dollar.  Sometimes  this  decimal  is  multiplied  by  1000,  the 
product  showing  the  number  of  dollars  taken  as  the  tax  on 
11000. 

390.  The  following  examples  illustrate  the  different  forms  in 
which  the  relation  of  tax  rate,  assessed  valuation,  and  amount 
of  taxes  appears  : 

1.  The  money  to  be  raised  by  tax  in  a  certain  town  is 
19000.  The  property  of  the  town  is  valued  at  1600,000. 
What  is  the  tax  rate  ? 

Statement  of  Relation :  of  $  600,000  =  $9000.     What  terms  of  relation 

are  given  ?    How  is  the  other  found  ? 


222  GRAMMAR  SCHOOL  ARITHMETIC 

2.  The  tax  rate  of  a  certain  county  is  .003  and  the  property 
is  valued  at  $24,567,800.  What  is  the  amount  of  the  tax 
budget  ? 

Statement  of  Relation:  .003  of  ^24,567,800  = .     How  is  the  required 

term  of  relation  found  ? 

3.  When  it  requires  a  tax  rate  of  .0132  to  raise  $264,000  in 
taxes,  what  is  the  valuation  of  the  property  taxed  ? 

Statement  of  Relation:  .0132  of =  $264,000.     How  may  the  required 

term  of  relation  be  found  ? 

391.     Oral 

1.  The  tax  budget  of  a  township  is  $12,000.  The  assessed 
valuation  of  the  property  in  the  township  is  $1,200,000. 

a.  What  is  the  tax  rate  ? 

b.  Mr.  A  has  property  in  this  township  assessed  at  $25,000. 
What  is  his  tax  ? 

c.  Mr.  B  pays  $15  taxes.  What  is  the  valuation  of  his 
property  ? 

2.  A  man's  city  taxes  were  $40  on  property  valued  at  $2000. 
What  was  the  tax  rate  ? 

3.  The  school  tax  in  a  village  having  property  to  the  amount 
of  $3,000,000  was  $9000. 

a.  What  was  the  school  tax  rate  ? 

b.  What  amount  did  a  man  pay  whose  property  was  assessed 
at  $15,000? 

c.  Mr.  Jones's  school  tax  was  $12.  What  was  the  valuation 
of  his  property  ? 

4.  The  tax  rate  of  a  certain  county  is  .0025.  The  tax 
budget  is  $75,000.     What  is  the  value  of  the  property  ? 

5.  A  town  has  352  citizens  who  pay  a  poll  tax  of  $1  apiece. 
The  entire  tax  budget  of  the  town  is  $5252. 

a.    How  much  money  must  be  raised  by  tax  on  the  property  ? 


TAXES  223 

h.  The  tax  rate  is  .007.  What  is  the  valuation  of  the 
property  ? 

c.  How  much  are  the  taxes  of  a  man  in  this  town,  who 
owns  property  assessed  at  $4000,  and  who  pays  one  poll  tax  ? 

6.  The  poll  tax  in  a  certain  town  is  $1.50,  and  there  are  400 
citizens  who  pay  poll  tax.  The  property  of  the  town  is  assessed 
at  $1,000,000,  and  the  rate  is  .01.  What  is  the  entire  amount 
raised  by  tax  ? 

7.  A  man's  property  is  assessed  at  $4000.  The  city  tax 
rate  is  .014,  the  county  rate  is  .004,  and  the  state  rate,  .002. 
The  poll  tax  is  $1.50.     What  is  this  man's  entire  tax? 

8.  If  the  rate  for  county  and  state  taxes  together  is  .005, 
what  is  my  bill  for  state  and  county  taxes  on  an  assessment  of 
$9000? 

9.  The  assessed  valuation  of  the  property  in  a  certain  county 
is  $70,000,000.  The  tax  rate  is  3  mills  on  a  dollar.  The 
county  has  an  income  from  various  sources  amounting  to 
$40,000.  After  collecting  all  the  taxes  and  other  income  and 
paying  all  the  expenses,  $5000  remains.  What  are  the  ex- 
penses of  the  county  ? 

10.  What  is  the  rate  when  $  24  will  pay  the  tax  on  property 
assessed  at  $1200? 

11.  What  is  the  tax  on  $10,000  of  property  when  the  rate  is 
.009345  ? 

12.  When  the  entire  budget  of  a  town  is  $35,000  and  500 
men  pay  $1  apiece  poll  tax,  how  much  must  be  assessed  on  the 
property  ? 

13.  When  the  tax  on  $1000  is  $18.57,  what  is  the  rate  per 
dollar  of  assessed  valuation  ? 

14.  $30  will  pay  the  tax  on  how  many  dollars'  worth  of 
property,  when  the  tax  rate  is  .015  ? 


224  GRAMMAR   SCHOOL   ARITHMETIC 

392.    Written 

1.  City  Tax  Budget  for  One  Year 

Interest $  49,755.44 

Comptroller 11,620 

City  Treasurer 18,450 

Department  Public  Instruction  (School  Funds)       ....  463,780 

Library  Fund 35,000 

Art  Museum 5,000 

Department  Charities  and  Correction 85,129 

Municipal  Lodging  House 4,071 

Veteran  Relief 8,000 

City  Engineer 35,959 

Public  Buildings  and  Grounds 16,000 

Department  Public  Works  (General  OflBce) 14,462 

Parks  and  Cemeteries 47,000 

Walks  and  Sidewalk  Repair 5,000 

Street  Cleaning 91,542 

Collecting  Garbage  and  Ashes 86,455 

Street  Repairs,  Sewers,  and  Bridges 64,120 

Municipal  Baths 4,000 

Public  Markets 3,382 

Lighting  Fund 114,000 

Boiler  Inspector 900 

Department  of  Law 13,720 

Municipal  Court 11,978 

Police  Court 6,000 

Department  of  Public  Safety  (General  Office) 7,520 

Police  Department 162,730 

Fire  Department 205,080 

Health  Department 55,925 

Department  of  Taxes  and  Assessments    .        .         .        .         .         .  19,200 

Executive  Department 8,400 

City  Clerk 9,000 

Civil  Service  Board           .         .       ' 2,600 

Election  and  Primary  Fund     .         .                                   ...  16,000 

Printing  and  Publishing  Fund          .     • 7,500 

Sealer  of  Weights  and  Measures 1,200 

Common  Council .        .         .  16,450 

Smoke  Inspector 1,200 

Plum  Street  Bridge 6,000 

Other  Expenses 139,879 

Total $ 

Less  Income  from  Licenses,  etc 246,228 

Net  Total $ 

From  the  above  city  tax  budget, 

a.    Find  the  total  expenses  of  the  city  for  the  year. 


TAXES  225 

5.    Find  the  net  total  of  the  tax  budget. 

c.  Find  the  tax  rate,  correct  to  four  places  of  decimals,  the 
assessed  valuation  of  the  real  property  in  the  city  being 
189,000,000  and  of  the  personal  property  19,000,000. 

d.  Find  the  amount  of  A's  city  tax  on  §15,000  of  personal 
property  and  $  5000  of  real  property. 

e.  In  this  city  the  county  and  state  taxes  are  paid  together, 
and  the  rate  is  .00363682.     What  is  A's  county  and  state  tax  ? 

/.  Mr.  B's  county  and  state  taxes,  computed  by  the  above 
rate,  amount  to  165.46276.  He  pays  165.47.  What  is  the 
valuation  of  his  property  ? 

g,  Mr.  C  owns  two  pieces  of  property  in  this  city,  one 
valued  at  1 600  and  the  other  at  13200.  What  is  the  entire 
amount  of  his  city,  county,  and  state  taxes  ? 

2.  The  valuation  of  property  in  a  certain  town  is  %  1,500,000, 
and  the  rate  is  |^  %.     What  is  the  tax  ? 

3.  The  tax  to  be  raised  in  a  certain  village  is  %  37,500. 
The  valuation  of  the  taxable  property  is  i  2,500,000. 

a.    What  is  the  rate  ? 

h.  What  will  be  A's  tax  on  1 15,000  real  estate,  and  $3000 
personal  property  ? 

c.  What  is  the  valuation  of  property  on  which  the  tax  is 
137.50? 

4.  The  property  of  a  town  is  assessed  at  $  1,250,000.  The 
tax  to  be  raised  is  %  15,975.  There  are  650  polls,  assessed  at 
%  1.50  each.  What  is  B's  entire  tax,  if  his  property  is  assessed 
at  %  2500,  and  he  pays  the  poll-tax  ? 

5.  The  officers  of  a  town  find  that  all  the  town  expenses 
for  a  year  will  amount  to  146,000.  The  tax-roll  shows 
real  estate  valued  at  %  2,000,000,  and  personal  property  at 
%  300,000.     What  is  the  tax  rate  ? 


226  GRAMMAR  SCHOOL  ARITHMETIC 

6.  The  tax  rate  in  a  certain  city  for  the  3^ear  1906  was 
$16.84  per  $1000  of  assessment.  The  city  treasurer  began 
to  receive  taxes  October  1,  and  taxpayers  who  failed  to  pay 
before  the  1st  of  November  had  a  one-per-cent  fee  added  to 
their  tax  bills.  What  was  the  tax  bill  of  Mr.  K,  whose  prop- 
erty was  assessed  at  i  7500  and  who  paid  his  taxes  on  the  .5th 
of  November  ? 

7.  If  the  assessed  valuation  of  a  village  is  12,384,564, 
and  there  are  750  polls  taxed  11.50  each,  what  must  be  the 
rate  of  taxation  to  meet  an  expense  of  $29,807.05? 

8.  A  sewer  was  built  in  a  street  980  feet  long,  at  a  cost  of 
$1999.20,  the  expense  being  assessed  to  the  owners  of  property 
on  each  side  of  the  street,  according  to  the  number  of  feet  of 
frontage  they  owned ;  that  is,  the  number  of  feet  their  land 
extended  along  the  street. 

a.   What  was  the  total  frontage  on  both  sides  of  the  street  ? 
h.    What  was  the  rate  per  front  foot  ? 

c.  What  was  the  sewer  tax  of  Mr.  M,  who  owned  one  lot 
4  rods  wide  and  another  50  feet  wide  ? 

EXCHANGE 

393.  A  draft  ^s  a  written  order  for  the  payment  of  money ^ 
made  in  one  place  and  payable  in  another. 

394.  A  bank  draft  is  an  order  made  by  a  bank  in  one  place^ 
directing  a  bank  in  a  different  place,  with  which  the  drawer  has 
funds  on  deposit,  to  pay  a  specified  sum  of  money  to  some  person, 
or  to  his  order,  or  to  the  bearer. 

395.  The  party  who  draws  a  draft  is  the  drawer;  the  party  to 
whom  the  order  is  addressed  is  the  drawee;  the  party  to  whom  a 
draft  is  payable  is  the  payee;  the  face  of  a  draft  is  the  sum 
ordered  to  be  paid. 


EXCHANGE 


227 


TO  THE  NATIONAL  PARK  BANK, 

NEW   YORK  CITY.   N.  Y. 


A  Bank  Draft 


In  the  draft  given  above,  the  drawer 
is  the  State  Bank  of  Utah,  of  which 
Henry  T.  McEwan  is  assistant  cashier; 
the  drawee  is  the  National  Park  Bank  of 
New  York,  and  the  payee  is  Henry  L. 
Fowler.  The  face  of  the  draft  is  1100. 
Observe  that  a  bank  draft  is  like  an 
ordinary  check,  except  that  both  the 
drawer  and  the  drawee  are  banks,  and  that  their  places  of 
business  are  in  different  cities  or  villages.  A  bank  draft  is 
sometimes  called  a  hanh  cheeky  because,  like  an  ordinary  check, 
it  is  an  order  drawn  by  one  party  upon  another  party,  with 
whom  the  first  party  has  fund§  deposited. 

396.  By  means  of  drafts,  payments  may  be  made  between 
different  places  without  actually  sending  the  money.  The 
method  of  making  such  payments  is  as  follows  : 

Let  us  suppose  that  Henry  L.  Fowler,  in  Salt  Lake  City, 
desires  to  send  to  Charles  Bryant,  at  Portland,  Me.,  f  100.  He 
goes  to  the  State  Bank  of  Utah,  in  Salt  Lake  City,  and  says 
to  the  teller  or  other  person  who  waits  upon  him,  "  I  wish  to 
buy  a  New  York  draft  for  $100,  payable  to  the  order  of  Henry 


228  GRAMMAR  SCHOOL  ARITHMETIC 

L.  Fowler."  (Some  banks  require  the  purchaser  of  a  draft  to 
fill  out  a  slip  with  the  name  of  the  payee  and  the  amount  of  the 
draft.)  The  teller  then  fills  out  and  hands  to  Mr.  Fowler  the 
draft  (page  227),  for  which  Mr.  Fowler  pays  1100  plus  a  small 
fee  to  pay  the  bank  for  its  services.  This  fee  is  called  the 
exchange.  The  exchange  is  sometimes  computed  at  a  certain 
per  cent  of  the  face  of  the  draft.     It  seldom  exceeds  |  %. 

Banks  often  sell  drafts  to  their  depositors  and  customers  with 
no  charge  for  exchange. 

Mr.  Fowler  indorses  the  draft  as  indicated  above,  incloses  it 
with  a  letter,  and  mails  it  to  Mr.  Bryant,  who  takes  it  to  a 
bank  in  Portland,  indorses  it  in  blank,  and  receives  f  100  for 
it.  The  transaction  is  complete  so  far  as  Mr.  Fowler  and 
Mr.  Bryant  are  concerned. 

Let  us  now  study  the  transaction  between  the  banks.  Every 
bank  of  importance  has  money  on  deposit  in  some  bank,  called 
its  correspondent,  in  one  or  more  of  the  great  money  centers 
of  the  country. 

The  National  Park  Bank  is  the  correspondent  of  the  State 
Bank  of  Utah.  The  bank  which  cashes  the  check  for 
Mr.  Bryant  in  Portland,  charges  $100  to  its  correspondent 
in  New  York  and  sends  the  draft  to  its  correspondent.  The 
correspondent  presents  the  draft  to  the  National  Park  Bank 
(through  the  clearing-house),  which  pays  $100  and  charges 
the  amount  to  the  State  Bank  of  Utah. 

Each  of  the  banks  has  now  received  and  paid  out  $100  in 
cash  or  credit ;  Mr.  Fowler,  in  Salt  Lake  City,  has  paid  out 
'$100,  and  Mr.  Bryant,  in  Portland,  has  received  $100  ;  and 
yet  no  money  has  actually  been  transferred  from  one  city  to 
the  other. 

Whenever  the  State  Bank  of  Utah  cashes  a  New  York  draft, 
it  sends   the   draft   to   its   correspondent   in   New   York   and 


EXCHANGE  229 

receives  credit  for  it,  which  is  the  same  as  sending  the  money 
received  for  drafts  which  it  has  sold. 

397.  In  New  York,  and  every  other  large  city,  many  checks 
and  drafts  are  received  by  one  bank,  payable  by  other  banks  in 
the  city.  For  the  sake  of  convenience,  all  these  checks  and 
drafts  are  sent  by  the  different  banks  to  one  place,  called  the 
clearing-house,  where  they  are  classified  and  sent  to  the  banks 
to  which  they  should  go,  and  balances  are  settled. 

398.  Making  payments  hy  means  of  drafts  or  money  orders  is 
exchange.     It  is  really  an  exchange  of  credits. 

399.  Exchange  between  places  in  the  same  country  is  domestic 
exchange. 

The  exchange  business  of  the  Middle  West  is  largely  carried 
on  through  Chicago  and  St.  Louis  banks.  A  similar  exchange 
business  is  conducted  between  every  great  money  center  and 
the  surrounding  section.  But  the  great  exchange  center  of 
the  United  States  is  New  York,  which  is  sometimes  called  the 
country's  clearing-house. 

400.  It  sometimes  happens  that  banks  in  one  city  have  large 
sums  on  deposit  with  banks  in  another  city,  and  need  currency 
for  immediate  use.  They  may  then  sell  drafts  at  a  discount 
from  their  face  value  in  order  to  get  the  money  at  once. 
When  the  balance  is  against  them,  they  may  sell  drafts  at  a 
premium,  which  is  a  certain  per  cent  above  their  face  value. 

401.  Personal  checks  are  used,  like  drafts,  in  making  pay- 
ments at  a  distance,  and  a  small  fee  for  collection  is  charged  by 
the  banks. 

402.  Oral  and  Written 

1.  Mr.  William  Harris,  in  South  Bend,  Ind.,  desires  to  send 
1200  to  his  nephew  Arthur  Otis,  who  is  in   college  in  New 


230  GRAMMAR  SCHOOL  ARITHMETIC 

Haven,  Conn.     How  much  will  a  New  York  draft  for  that 
sum  cost,  if  the  exchange  is  -^^  %  ? 

2.  The  banks  making  the  above  exchange  are  the  Farmers' 
Bank  of  South  Bend  and  its  correspondent,  the  Marine  Bank 
of  New  York,  the  Exchange  National  Bank  of  New  Haven  and 
its  correspondent,  the  Industrial  Bank  of  New  York.  Describe 
the  entire  transaction. 

3.  Write  the  draft,  and  indorse  it  properly. 

4.  Minneapolis  banks  have  large  balances  in  New  York 
banks.  Therefore  they  are  selling  New  York  drafts  at  gV  % 
discount. 

a.  What  is  the  cost  in  Minneapolis  of  a  New  York  draft  for 
1800? 

Hint.  — $800  -  i^%  of  |800  =  ? 

h.  Write  the  draft  in  question  a,  the  parties  being  James  B. 
Weaver,  the  Produce  Exchange  Bank  of  New  York,  and  the 
Minnehaha  National  Bank  of  Minneapolis. 

e.  Charles  O.  Richards  of  Minneapolis  has  collected  $  3938.03 
for  John  Howe  &  Co.  of  Scranton,  Pa.  Write  the  New  York 
draft  that  he  can  purchase  with  that  sum  at  the  Minnehaha 
National  Bank. 

Statement  of  Relation :  99^1%  of =  $3938.03. 

5.  Milwaukee  banks  have  small  balances  in  New  York  banks. 
They  are  selling  New  York  exchange  at  ^  %  premium. 

a.    What  is  the  exchange  on  a  New  York  draft  for  17500  ? 

6.  The  exchange  on  a  draft  sold  to  Cyrus  Johnson  by  the 
Northeastern  Bank  of  Milwaukee  was  $20.50.  What  was  the 
face  of  the  draft  ? 

Statement  of  Relation  :  J%  of =  $  20.50. 

c.  Write  the  draft  in  question  5,  making  the  Traders'  Bank 
of  New  York  the  drawee. 


COMMERCIAL  DRAFTS  231 

6.  What  is  the  rate  of  exchange  when  a  draft  for  17500 
costs  17505? 

Statement  of  Relation: %  of  $7500  =  $5. 

7.  The  discount  on  a  draft  for  $8400  is  §7.  What  is  the 
rate  of  discount  ? 

Statement  of  Relation :  %  of  $  8400  =  $  7. 

8.  When  money  was  scarce  in  San  Francisco,  and  large 
balances  were  held  in  Chicago,  a  man  in  San  Francisco  bought 
a  Chicago  draft  of  -f  12,800,  paying  1 12,784  for  it.  At  what 
rate  of  discount  did  he  buy  the  draft  ? 

COMMERCIAL  DRAFTS 

403.  Drafts  are  frequently  used  as  a  means  of  collecting 
bills.  For  example,  Horace  Prang  of  Columbus,  O.,  owes 
Loetzer  &  Co.  of  Buffalo,  an  account  of  1500,  payable  Aug.  26, 
1908.     Loetzer  &  Co.  make  out  the  following: 

Time  Draft 


jRlJg  I  jU/i^ffh,  i'Lx^t^  .a.4^h/f^Uxctt^  Pay  towe  order  of 

a^pS  -    WucTecGived  and  ebai^e  to  account  nf 

§1 1  ^r^k^'^^X^i  c^e^^^^ 


Loetzer  &  Co.  deposit  this  draft  in  the  Bank  of  Buffalo, 
which  sends  it  to  some  bank  in  Columbus.  This  bank  presents 
the  draft  to  Horace  Prang,  who,  if  he  is  willing,  writes  in  red 
ink  across  its  face,  "Accepted,  July  1,  1908"  (if  that  is  the 


232  GRAMMAR  SCHOOL   ARITHMETIC 

day  on  which  it  is  presented)  and  signs  his  name.  The  draft 
is  now  equivalent  to  Mr.  Prang's  bank  note,  payable  Aug.  26, 
indorsed  by  Loetzer  &  Co.  It  is  returned  to  the  Bank  of 
Buffalo,  which  will  discount  it  at  once,  if  Loetzer  &  Co.  are 
customers  in  good  standing,  and  credit  them  with  the  pro- 
ceeds, less  a  small  fee  for  collection. 

If  the  draft  were  an  order  to  pay  "sixty  days  after  sight,'' 
and  accepted  by  Mr.  Prang,  he  would  be  entitled  to  sixty  days, 
after  its  presentation  and  acceptance,  before  paying  it.  If  not 
paid  then,  it  would  be  protested,  like  a  bank  note. 

404.  Shippers  often  use  drafts  as  a  means  of  collecting  pay- 
ment for  goods  on  delivery,  or  of  securing  promise  of  payment 
at  a  specified  time. 

Suppose  the  Empire  Elevator  Company  of  Buffalo  is  sending 
a  carload  of  corn,  containing  700  bushels,  billed  at  60  cents  a 
bushel,  to  the  Smith  Milling  Company  of  Springfield,  Mass. 
The  Elevator  Company  receives  a  bill  of  lading  from  the  rail- 
road company,  which  the  Smith  Milling  Company  must  have 
before  it  can  get  permission  to  take  the  corn  from  the  car  at 
Springfield.  The  Elevator  Company  takes  this  bill  and  deposits 
it  in  the  Marine  National  Bank  of  Buffalo  together  with  the 
following  draft : 


111 

oil 


$  Jf-Zo.—  Buffalo.  N,  Y..   A^^^.  ^       190  •T 


.PAY  TO  THE  ORDER  OF  THE 


IVIarine  National  Bank  of  Buffalo, 

;^rT4>t^/?2**-^^*^  -*<*-!^'2SJ^««*^^^^r  — —  ■"  'Dollars. 

VALUE  received  AND  CHARGE  TO  THE  ACCOUNT  OF 
TO  JL/fi^^^^Mi^^(^ 


:}  v^^^^i^^^^^^^^ 


COMMERCIAL  DRAFTS  233 

The  Marine  National  Bank  of  Buffalo  sends  the  draft  and 
bill  of  lading  to  a  bank  in  Springfield,  which  presents  it  to  the 
Milling  Company  for  payment.  If  the  Milling  Company  pays 
the  draft,  it  receives  the  bill  of  lading,  which  entitles  it  to  take 
the  corn  from  the  car.  The  Springfield  bank  remits  the 
amount,  by  draft  or  otherwise,  to  the  Marine  National  Bank  of 
Buffalo,  which  credits  it  to  the  Empire  Elevator  Company,  less 
the  cost  of  collection. 

If  the  Smith  Milling  Company  refuses  to  pay  the  draft,  the 
Springfield  bank  notifies  the  Marine  National  Bank  of  Buffalo, 
which  notifies  the  Empire  Elevator  Company.  Then  the  Ele- 
vator Company  mast  arrange  to  have  the  corn  returned  or  dis- 
posed of  in  some  other  way. 

405.  When  the  drawee  has  accepted  a  drafts  he  is  called  the 
acceptor  and  the  draft  is  called  an  acceptance. 

406.  When  a  draft  is  drawn  payable  a  certain  number  of 
days  or  months  after  sights  it  is  necessary  to  have  a  date  in  the 
acceptance  so  as  to  determine  the  day  of  maturity  of  the  draft. 

A  draft  drawn  payable  after  date  may  be  properly  accepted 
by  the  mere  signature  of  the  drawee  across  its  face,  though  the 
date  is  also  desirable. 

407.  A  draft  payable  at  sight  (i.e.  at  the  time  of  presentation) 
is  called  a  sight  draft  ;  a  draft  payable  at  a  specified  time  after 
sight  or  after  date  is  called  a  time  draft. 

408.  A  sight  draft  may  be  accepted  payable  a  certain  time 
after  date  of  acceptance.  It  then  has  the  force  of  a  note  and 
may  be  discounted  like  a  time  draft  or  bank  note. 

409.  The  discount  on  a  time  draft,  and  the  cost  of  collection^ 
called  the  exchange^  of  any  draft  are  computed  on  the  face  of 
the  draft. 


234  GRAMMAR  SCHOOL   ARITHMETIC 

410.  The  face  of  a  draft  less  the  exchange  and  the  discount 
(on  a  time  draft)  is  called  the  net  proceeds. 

Note.  —  In  states  where  grace  is  allowed  by  law,  add  three  days  to  the 
time  in  finding  the  maturity  and  term  of  discount. 

411.  Written 

1.   What  was  the  exchange  for  collecting  the  draft  on  the 
Smith  Milling  Company,  page  232,  at  -^^  % 


9 


2.  a.  Compute  the  discount  on  the  draft  accepted  by  Horace 
Prang,  page  231,  from  the  day  of  acceptance  to  the  day  of 
maturity  at  6%. 

h.  If  the  charge  for  collection  was  yo%'  what  sum  was 
credited  to  the  account  of  Loetzer  &  Co.,  by  the  Bank  of 
Buffalo  ? 

3.  What  is  the  cost  of  collecting  the  following  draft  at  |  %  ? 


4t€> 


^^ni3L 


'  X^t^z^/j^i^VT^ 


4.  William  H.  Warner  of  Burlington,  la.,  draws  on  H.  H. 
Franklin  of  Dubuque,  for  f  1800,  at  60  days  sight,  through  the 
National  Bank  of  Burlington. 

a.  Write  the  draft. 

h.  Compute  the  discount,  the  draft  having  been  accepted, 
and  discounted  at  6  %  on  the  day  of  acceptance. 


COMMERCIAL  DRAFTS  235 

c.  If  the  exchange  for  collection  was  y^  %,  what  were  the 
net  proceeds  of  the  draft  ? 

5.  Find  the  net  proceeds  of  the  following  draft  on  the  day 
of  acceptance,  computing  discount  at  5%  per  annum  and 
exchange  at  |^%. 


Value  nxmvd  anddkarye  io  account  of 


6.  Find  the  net  proceeds  of  a  draft  for  $1440  payable  60 
days  after  date,  the  rate  of  discount  being  7  %  per  annum,  the 
day  of  discount  thirty  days  after  date,  and  the  exchange  |^%. 

7.  The  net  proceeds  of  a  60-day  draft  discounted  at  6  %  on 
the  day  of  date,  exchange  at  J%,  were  $1977.50.  What  was 
the  face  of  the  draft  ? 

EXCHANGE  BY  POSTAL  MONEY  ORDER 

The  Post  Office  Department  offers  a  convenient  method  of 
exchange,  for  small  amounts,  in  the  form  of  postal  money  orders. 

412.  A  postal  money  order  is  a  written  agreement,  signed  hy 
the  postmaster  of  a  certain  post  office.,  that  the  postmaster  of 
another  post  office  will  pay  a  specified  sum  of  money  to  the  person 
named  in  the  order. 


236  GRAMMAR   SCHOOL   ARITHMETIC 

The  following  form  shows  the  essential  parts  of  a 
Postal  Money  Order 


[Name  of 

office  issuing  the  order] 
DATE 

NO. 

THE    POSTMASTER 

AT     [Name 

of  office   on  which   order  is   drawn] 

WILL    PAY 

THE    SUM    OF 

DOLLARS 

CENTS 

P 

words  for  dollars 

DOLLARS 

P 

figures  for  cents 

CENTS 

TO 

THE 

ORDER    OF  [Name  of 

person  to  whom  order  is  payable] 

NAME  OF  REMITTER 

[Signature 

of]    POSTMASTER 

413.  These  orders  may  be  purchased  at  any  money  order  post 
office.  All  except  the  smaller  village  and  rural  post  offices  are 
money  order  offices. 

The  purchaser  (called  "the  remitter,"  in  the  order)  incloses  the  order  in 
an  envelope,  and  mails  it  to  the  payee  named  in  the  order.  The  payee 
takes  it  to  the  post  office  named  in  the  order,  where  he  receives  in  cash  the 
face  value  of  the  order. 

All  money  order  offices  sell  postal  money  orders  (for  amounts  not 
exceeding  $100),  payable  at  money  order  offices  in  this  country  or  in  foreign 
countries. 

The  following  table  shows  the  fees  that  must  be  paid,  in  addition  to  the 
face,  for  postal  money  orders  payable  in  the  United  States: 
Face  of  Order 

$2.50  or  less 

Over  2.50  and  not  exceeding 
Over  5.00  and  not  exceeding 
Over  10.00  and  not  exceeding 
Over  20.00  and  not  exceeding 
Over  30.00  and  not  exceeding 
Over  40.00  and  not  exceeding 
Over  50.00  and  not  exceeding 
Over  60.00  and  not  exceeding 
Over  75.00  and  not  exceeding 
Fees  for  foreign  orders  are  about  three  times  as  great  as  for  domestic, 
ranging  from  10)^5  to  $1.00. 


Fee 

.   .     S^ 

$5.00    . 

.   .     5;* 

10.00    . 

.     Sf 

20.00    . 

.    lOj^ 

30.00    . 

.    12^ 

40.00    .    . 

.    15;^ 

50.00    .    . 

.    18^ 

60.00    .    . 

.   20^ 

75.00    .    . 

.   25^ 

100.00    .    . 

.    SOf 

MONEY  ORDERS 


237 


414.    Oral 

Using  the  table  of  rates  on  page  236, 

1.  Find  the  total  cost  of  a  postal  money  order  for — 

a.  13.00  d.  143.25  g,  $86.31  j.  128.98 
h.  $4.28  e,  189.41  h.  $72.05  k,  $90.89 
c,    $1.75        /.    $99.99         ^.    $50.10        h    $88.95 

2.  Find  the  cost  of  two  postal  money  orders  which,  together, 
will  pay  a  bill  of  $137.55  at  Wanamaker's  store  in  Philadel- 
phia. 

3.  Find  the  cost  of  postal  money  orders  sufficient  to  pay 
a  bill  of  $500. 

Make  and  solve  other  problems. 


EXCHANGE   BY   EXPRESS   MONEY   ORDER 

415.  An  express  money  order  is  a  written  agreement  hy  an 
express  company  to  pay  to  the  order  of  a  person  named  in  the 
order  a  specified  sum  of  money. 

The  following  is  the  usual  form: 


When  CouNTERsrcNEO 

BY  AOCNT  AT  POINT  Or  ISSOC 


7-5997858 


JJotiliniratal  ^p^ss  JJmnpairo 


TOTRANSMfTANO 


PAyiO  THE  ORDER  OF. 

The  Sum  OF    CL/^VXjty>^   d^^^-^^ts^^-^ 

0>ON' 


-  TooDOLLARS 


Issued  at 


t^2:   Stat,  or      C/^, £.    ,;^^*«  ■"  "';";i      .    /7 

MUTILATION  or  THIS  OROCft  RCMOCRS  IT  VOID^  \^^ 


FEB.&l  1908 


ANY  cnASURC.M.TCRATiOM,OerACCMCNT  OR  I 


238  GRAMMAR  SCHOOL  ARITHMETIC 

The  fee  is  the  same  as  that  for  issuing  a  postal  money  order 
for  the  same  amount.  It  is  called  the  exchange  for  issuing 
the  order. 

416.  An  express  money  order  is  negotiable  and  can  he  trans- 
ferred hy  indorsement^  like  a  check  or  hank  draft. 

417.  An  express  money  order,  issued  by  any  express  com- 
pany, will  be  cashed  for  its  full  face  value  at  any  of  the  com- 
pany's offices  in  this  country,  or  by  any  other  express  company, 
or  by  any  bank. 

418.  Oral 

1.  Name  two  similarities  between  the  method  of  exchange 
by  express  money  order  and  that  by  postal  money  order. 

2.  Name  two  differences. 

3.  What  must  be  paid  in  Latrobe,  Pa.,  for  an  express  money 
order  large  enough  to  pay  a  bill  of  127.27  in  Los  Angeles,  Cal.? 

Make  and  solve  other  problems. 

EXCHANGE  BY  TELEGRAPH  MONEY  ORDER 

419.  Exchange  by  telegraph  money  order  is  more  expensive 
than  that  by  express  or  postal  money  order.  It  is  used  only  in 
cases  of  emergency,  when  credits  must  be  transmitted  without 
loss  of  time,  so  that  money  paid  in  one  place  may  be  instantly 
available  in  another  place  at  some  distance  from  the  first.  ^ 

The  method  by  which  this  form  of  exchange  is  made  is  as 
follows : 

The  person  desiring  to  remit  money  goes  to  a  telegraph  office  and  pays 
the  money  to  the  person  who  attends  to  that  branch  of  the  business.  A 
message  is  then  sent,  directing  the  telegraph  office,  at  the  place  where  the 
money  is  wanted,  to  pay  the  amount  to  the  person  designated.     Before  receiv- 


FOREIGN   EXCHANGE  239 

ing  the  money,  that  person  is  required  to  satisfy  the  representatives  of  the 
telegraph  company,  by  identification  or  otherwise,  that  he  is  the  person 
to  whom  the  money  is  directed  to  be  paid. 

420.  The  present  rate  for  telegraph  money  orders  is  twice 
the  cost  of  a  ten-word  message^  plus  one  per  cent  of  the  amount  of 
the  order.  If  the  amount  of  the  order  is  less  than  f  25,  the  fee  is 
the  same  as  if  the  order  were  for 


421.  Oral 

1.  What  is  the  cost  in  Syracuse,  N.Y.,  of  a  telegraph  money 
order  for  |75,  payable  in  Atlanta,  Ga.,  the  cost  of  a  ten-word 
message  being  60^? 

2.  What  is  the  cost  in  Scranton,  Pa.,  of  a  telegraph  money 
order  for  $50,  payable  in  San  Francisco,  the  rate  for  a  ten- 
word  message  being  fl.OO  ? 

3.  What  is  the  cost  in  Utica  of  a  telegraph  money  order 
for  $100,  payable  in  Harrisburg,  Pa.,  the  rate  for  a  ten- word 
message  being  40)^? 

Make  and  solve  other  problems. 

FOREIGN  EXCHANGE 

422.  Exchange  between  places  in  different  countries  is  for- 
eign exchange. 

423.  The  principal  gold  coin  of  Great  Britain  is  the  sovereign, 
equal  to   XI.     It  is  equiva- 
lent to  $4.8665. 

Which  of  our  coins  is  most 
nearly  like  the  sovereign  ? 

The  shilling  is  a  silver  coin 
equal  to  -^-^  of  a  pound,  ster- 


240 


GRAMMAR  SCHOOL   ARITHMETIC 


ling.  It  is  equivalent  to 
about  how  many  cents  ?  It 
closely  resembles  what  Amer- 
ican coin  ? 

The    English  penny  is  -^^ 
of  a  shilling.     It  is  equiva- 
lent to  about  how  many  cents  in  our  money  ? 

424.   The  principal  gold  coin  of  France  is  the  20-franc  piece, 
nearly  equivalent  to  $4  of  our  money. 

The  franc  is  a  silver  coin 
equivalent  to  $.193  of  our 
money.  What  American 
coin  is  most  nearly  like  the 
franc?  The  franc  is  also 
the  principal  coin  of  Belgium 
and  Switzerland. 

The  lira  (plural  lire)  of  Italy  has  the  same 
value,  and  is  exchanged  evenly  for  the  franc. 
A  dollar  is  equivalent  to  about  how  many 
francs  ?     How  many  lire  ?     How  many  shil- 
lings ? 

The  principal  coin  of  Germany  is  the 
coin        y^^^^^^^^^^^^^^^^:::^^ 


425. 

mark.     It    is   a   silver 

equivalent  to  $.238.     What 

American  coin  is  most  nearly   //  V 

like  the  mark  ?     One  dollar  \|         ^  ^^        I 


is  equivalent  to  about  how 
many  marks  ?  Four  marks 
are  equivalent  to  how  many  cents  ? 

426.   Foreign   drafts,    called   bills   of   exchange,    are   always 
expressed   in   the   money  of  the  country  in  which   they   are 


FOREIGN  EXCHANGE 


241 


payable.  Sterling  bills  (drafts  on  Great  Britain  and  Ireland) 
are  expressed  in  pounds,  shillings,  and  pence  ;  drafts  on  France, 
Belgium,  and  Switzerland  in  francs;  on  Italy,  in  lire;  and  on 
Germany,  in  nciarks  (re icli marks). 

427.  Foreign  drafts  are  usually  issued  in  sets  of  two,  known 
as  the  first  and  second  of  exchange.  When  either  of  them  is 
paid,  the  other  becomes  void. 


Set  of  Exchange 


Exchange  for  £  260-^-^ 

J{ew  York,  ^£.L  26,  1907 

At  sight  of  this._.jUQ,t___of  exchange  (second  unpaid) 

pay  to  the  order  of Ra{>-&vt  /71ci(S.^an.cild 

Value  received,  and  charge  to  the  account  of 

To  £o-m.{^c^vcl  y  (^a. 
JVo.  <?f  ^       £andoft, 


JSva-w-yv  JSvotk&x^  V^  (^a. 


The  second  draft  of  the  set  is  like  the  first,  except  the  inter- 
change of  the  words  first  and  second. 

FOREIGN  EXCHANGE   QUOTATIONS 

428.  The  values  given  on  pages  239  and  240  for  various 
coins  are  the  exact  equivalents  of  those  coins,  in  our  money. 
This  is  called  the  intrinsic  par  of  exchange.  The  exchange 
values  of  those  denominations,  however,  fluctuate  from  day  to 
day,  like  the  prices  of  corn,  wheat,  and  cotton,  and  are  quoted 
in  the  daily  papers. 


242  GRAMMAR  SCHOOL   ARITHMETIC 

429.  Exchange  on  Great  Britain  and  Ireland  is  quoted  at 
the  number  of  dollars  that  must  be  paid  for  one  pound  of  ex- 
change;  e.g.  "Exchange  on  Liverpool,  4.87|-,"  means  that  a 
draft  on  Liverpool  costs  at  the  rate  of  $4. 87 J  for  every  pound 
of  its  face.  Time  drafts  are  quoted  at  a  lower  rate  than  sight 
drafts,  because  the  drawer,  who  sells  the  draft,  has  the  use  of 
the  money  until  the  draft  matures.  There  is  no  discount  to  be 
computed,  because  that  is  accounted  for  in  the  quotation. 

"London  sight  4.86|;  60  days,  4. 85 J"  means  that  London 
sight  drafts  are  sold  at  i4.86|  per  pound  of  their  face,  while 
60-day  London  drafts  are  sold  at  f  4.85|  per  pound  of  their 
face. 

Exchange  on  France,  Belgium,  and  Switzerland  is  quoted  at 
the  numher  of  franca  of  exchange  that  may  be  bought  for  one 
dollar 'i  e.g.  "Exchange  on  Brussels,  5.17|-"  means  that  one 
dollar  must  be  paid  for  every  5. 17 J  francs  of  the  face  of  the 
draft. 

Exchange  on  Italy  is  quoted  at  the  numher  of  lire  of  exchange 
that  can  be  bought  for  one  dollar. 

Exchange  on  Germany  is  quoted  at  the  numher  of  cents  that 
must  be  paid  iorfour  marks  of  the  face  of  the  draft;  e.g.  "Ex- 
change on  Hamburg,  96|^,"  means  that  f  .96|-  must  be  paid  for 
every  four  marks  of  the  face  of  the  draft. 

The  cost  of  cable  transfers,  or  telegraph  money  orders  to  for- 
eign countries,  is  computed  by  finding  the  exchange  value  of 
the  order  according  to  the  quotation,  and  adding  thereto  a 
certain  per  cent,  usually  from  \%  to  1%,  plus  the  cost  of  the 
cable  message. 

Post  remittances,  or  foreign  postal  money  orders^  are  sold  at 
their  exchange  value,  plus  a  certain  fee  for  issuing  and  paying 
the  order. 

Thus  it  will  be  seen  that  no  general  rule  can  be  given  for 


FOREIGN  EXCHANGE  243 

computation  of  foreign  exchange.  The  first  step  in  every  prob- 
lem, however,  is  to  determine  what  the  quotation  means,  accord- 
ing to  the  explanation  given  above.  When  that  is  determined 
and  clearly  fixed  in  the  mind,  the  student's  experience  in  apply- 
ing principles  to  given  conditions  so  as  to  secure  correct  results 
should  enable  him  to  solve  the  problem. 

430.    Oral 

1.  Give  the  meaning  of  each  of  the  following  quotations  of 
exchange : 

a.  London  4.87^  I.   Munich  95| 

b.  Edinburgh  4.86f  m.   Manchester  4.86f 

-      5  sight  4.87  n.   Frankfort  97 

c.  -^-ondon -J  g^  ^^^^  ^  g^^  ^^   Zurich  5.16^ 

d.  Paris  5.16J  p.  Berne  5.17^ 

e.  Geneva  5.18  q.  Belfast  4.87f 
/  Milan  5.17  r.  Florence  5.14f 
g.  Antwerp  5.17  s.  Bremen  87f 

h.  Berlin  95^  \  sight  4.86 

i.  Glasgow  4.87^  ^'   ■^'''^''P''°^  ^  60  days  4.83 

J.  Dresden  96  u.   Brussels  5.19 

k.  Rome  5.18J  v.   Hamburg  96| 

2.  When  four  marks  cost  96  cents,  what  is  the  cost  of  1 
mark  ?     Of  100  marks  ? 

3.  What  must  be  paid  for  a  draft  for  £  100,  when  one  pound 
of  exchange  costs  $4.87  ? 

4.  When  exchange  on  Paris  is  at  the  rate  of  5.19  francs  for 
$1,  what  is  the  face  of  a  draft  that  may  be  bought  for  $10  ? 

5.  When  exchange  on  France  is  quoted  at  5.20,  how  may 
we  find  the  cost  of  one  franc  ? 

6.  What  is  the  cost  of  a  draft  for  519  francs  when  5.19 
francs  of  exchange  may  be  bought  for  $1? 


244  GRAMMAR   SCHOOL   ARITHMETIC 

7.  When  a  draft  on  Liverpool  for  XIOOO  costs  $4875,  what 
is  the  rate  of  exchange  ?     (What  will  £  1  cost  ?) 

8.  What  is  the  rate  of  exchange  when  a  draft  for  100  marks 
costs  $24  ?     (What  will  4  marks  cost  ?) 

9.  What  is  the  rate    of   exchange  when  a  draft  for  1038 
francs  costs  $200  ?     (How  many  francs  will  f  1  buy?) 

10.  In  what  denomination  are  drafts  on  England  expressed  ? 
On  France  ?  On  Scotland  ?  On  Belgium  ?  On  Germany  ? 
On  Italy  ?     On  Switzerland  ?     On  the  United  States  ? 

11.  What  is  the  rate  of  exchange  when  1 980  will  buy  a 
draft  for  £  200  ? 

12.  At  what  rate  of  exchange  will  f  50  buy  a  draft  for  258 
francs  ? 

13.  At  what  rate  of  exchange  will  a  draft  on  Rome  for  1040 
lire  cost  $200? 

14.  When  the  rate  of  exchange  on  Germany  is  98,  what  is  the 
cost  per  mark  of  a  draft  on  Cologne  ? 

15.  What  is  the  cost  per  franc  of  a  draft  on  Brussels,  when 
the  rate  of  exchange  is  5.00  ? 

431.    Written 

1.  a.   When  exchange  on  London  is  quoted  at  4.87,  what  is 
the  cost  of  a  London  draft  for  £  250  ? 

h.  What  is  the  face  of  a  London  draft  that  can  be  bought 
for  $4383? 

2.  a.  What  is  the  cost  of  a  draft  on  Paris  for  2069.5  francs 
when  exchange  is  quoted  at  5. 17 J  ? 

b.  What  is  the  face  of  a  draft  on  Berne  that  can  be  bought 
for  $689  when  the  rate  of  exchange  is  5.18  ? 

3.  A  Liverpool  draft  for  £850  cost  $4160.75.     What  was 
the  rate  of  exchange  ? 


FOREIGN  EXCHANGE  245 

4.  When  1240  will  buy  a  draft  for  1242.60  francs,  how  many 
francs  of  exchange  will  $  1  buy  ?  What  is  the  rate  of  exchange  ? 

5.  When  the  quotation  for  London  exchange  is  "  sight  4.  S7^ ; 
30  da.  4.86,"  what  is  the  difference  between  the  cost  of  a  sight 
draft  for  £  470  and  a  30-day  draft  for  the  same  amount  ? 

6.  What  is  the  cost  of  a  draft  on  Berlin  for  948  marks  when 
exchange  is  quoted  at  98  ? 

4  marks  cost  $.98 

1  mark  cost  ^* — 
4 

948  marks  cost   ^—  x  —  (cancel) 

7.  A  bill  of  exchange  on  Frankfort  for  8000  marks  costs  how 
much  when  the  rate  of  exchange  is  .95f  ? 

8.  What  is  the  face  of  a  bill  of  exchange  on  Bremen  that  can 
be  bought  for  $2031.75  when  the  rate  is  96|? 

9.  Brown  Brothers  &  Co.,  bankers,  sent  out  to  their  cor- 
respondents, Oct.  14,  1907,  the  following  list  of  quotations : 

NO.  447.      BROWN  BROTHERS  &  CO.        new  yobk,  Oct.  14, 1907 

ENGLAND,  SCOTLANDlc  IRELAND  .    •     $4.87  \  per  Pound 

LONDON  &  LIVERPOOL  ONLY 4.86U      Sterling 

FRANCE,  BELGIUM  &  SWITZERLAND     •    5.16 

PARIS  ONLY 5.161  I    Francs 


1 


ANTWERP  &  BRUSSELS  ONLY 5.17     per  Dollar 

ZURICH,  ST.  GALL,  BASLE,  BERNE&GENEVAONLY5.I6J  '  '^ 

ITALY .    6.161)        Lire 

GENOA,   MILAN,  NAPLES,  FLORENCE,  LUCCA  & 
ROME  ONLY      5.l4i 

GERMANY :95V] 

BERLIN,  BREMEN,  CHEMNITZ,  DRESDEN.  FRANK-  I       CtS.  per 

FORT,    HAMBURG,    HANOVER,    MANNHEIM,    MU-  (Four  Marks 

NICH  &  NUREMBERG  ONLY 95  J 

HOLLAND .4042)    cts.  per 

AMSTERDAM  ONLY 40?®  i    Guilder 


246  GRAMMAR  SCHOOL  ARITHMETIC 


AUSTRIA  86  HUNGARY 203°]    cts. 


per 


VIENNA,  BUDAPEST,  AUSSIG,  PRAGUE,  TEPLITZ  h       i^^^^^ 

&  TRIESTE  ONLY 2025  j      rs.rone 

NORWAY,  SWEDEN,  DENMARK&  ICELAND  .26^1  qx 

CHRISTIANIA,  COPENHAGEN,  GOTEBORG&STOCK-  [   i/^'Jfr.^ 

HOLM  ONLY 2665J  i^ronor 

FINLAND .1945)   cts.  per 

HELSiNGFORS  &  wiBORG  ONLY 1919)   pinmark 

RUSSIA .611)    cts.  per 

ST.    PETERSBURG,    MOSCOW,    ODESSA,    BAKU,  \       D„hlo 

CHARKOFF  &  KIEF  ONLY 5liJ       riuaie 

POST  REMITTANCES  at  the  ordinary  rate  plus  15c.  per  payment.     CABLE  TRANSFERS  h%  higher  plus  cost 

of  message. 

By  these  quotations,  find: 

a.  The  cost  of  a  set  of  exchange  on  Antwerp  for  13,442 
francs. 

h.  The  face  of  a  London  bill  that  can  be  purchased  for 
11459.50. 

c.  The  cost  of  a  bill  of  exchange  on  Dublin  for  £  420  10s. 

d.  The  cost  of  a  set  of  exchange  on  Glasgow  for  £  200  10s.  6 J. 

e.  The  face  of  a  Berlin  draft  that  can  be  bought  for  $  190. 
/.    The  cost  of  a  set  of  exchange  on  Rome  for  20,575  lire. 

ff.  The  cost  of  a  cable  transfer  of  20,650  francs  to  Paris, 
computing  exchange  at  the  quoted  rate,  then  adding  -^  %  plus 
the  cost  of  the  cable  message,  which  was  $5.04. 

h.  The  face  of  a  London  bill  that  can  be  bought  for 
$2189.25. 

t.  The  face  of  a  bill  of  exchange  on  Geneva  that  can  be 
bought  for  1200. 

y.    The  cost  of  a  Dresden  bill  of  exchange  for  896  marks. 

10.  A  man  has  saved  $400,  and  desires  to  send  it  to  his 
family  in  Naples.  What  is  the  face  of  the  draft  which  he  can 
buy  with  that  sum,  by  the  above  quotation  ? 


THE  METRIC   SYSTEM  247 

11.  Make  and  solve  problems  of  exchange  on  other  countries 
named  in  the  list  of  quotations. 

THE  METRIC   SYSTEM 

The  metric  system  of  weights  and  measures  is  a  decimal 
system  which  originated  in  France  a  little  more  than  one  hun- 
dred years  ago.  It  is  the  legal  sj^^stem  in  most  of  the  civilized 
world  except  Great  Britain  and  the  United  States. 

In  our  own  country,  it  is  used  in  the  sciences  and  in  some 
branches  of  the  government  business. 

Being  a  decimal  system,  it  is  much  simpler  than  the  English 
system  which  we  use  ;  for  all  reductions  from  one  denomination 
to  another  may  be  made  simply  by  moving  the  decimal  point. 

LINEAR  MEASURE 

432.  The  standard  unit  of  linear  measure  in  the  metric 
system  is  the  meter.  It  is  determined  by  taking  one  ten- 
millionth  part  (very  nearly)  of  the  distance  from  the  earth's 
equator  to  either  of  its  poles,  measured  on  a  meridian.  It  is 
equal  to  39.37  inches. 

433.  Oral 

1.  What  denomination  in  the  English  linear  measure  is  most 
nearly  like  the  meter  ? 

2.  Draw  a  line  one  meter  long. 

3.  Hold  your  hands  one  meter  apart. 

4.  A  meter  is  about  how  many  feet  long  ? 

5.  How  many  meters  long  is  your  schoolroom  ?     Wide  ? 
High? 

6.  About  how  many  meters  are  there  in  a  rod  ? 


248  GRAMMAR  SCHOOL   ARITHMETIC 

7.  About  how  many  meters  long  is  a  rifle-range  whose 
length  is  500  yd.  ? 

8.  Your  height  is  about  how  many  meters  ? 

9.  How  many  meters  high  can  you  reach  on  the  blackboard  ? 

How  the  Table  is  made 

434.  Divide  a  meter  into  ten  equal  parts.  One  of  these 
parts  is  a  decimeter.  Dec  is  a  Latin  prefix  meaning  tenth. 
About  how  many  inches  long  is  a  decimeter  ?  Show  with  your 
hands  the  length  of  a  decimeter.  What  part  of  a  meter  is  a 
decimeter  ? 

Divide  a  decimeter  into  ten  equal  parts.  One  of  these  parts 
is  a  centimeter.  Cent  is  a  Latin  prefix  meaning  hundredth. 
What  part  of  an  inch  is  a  centimeter  ?  Show  its  length.  How 
many  centimeters  in  one  meter  ?  What  part  of  a  meter  is  a 
centimeter  ? 

Divide  a  centimeter  into  ten  equal  parts.  One  of  these  parts 
is  a  millimeter.  Mill  is  a  Latin  prefix  meaning  thousandth. 
What  part  of  a  meter  is  a  millimeter  ?  How  many  millimeters 
in  a  meter  ?     What  part  of  an  inch  is  a  millimeter  ? 

Ten  meters  make  one  dekameter.  Beha  is  a  Greek  prefix 
meaning  ten.  How  many  rods  in  a  dekameter?  How  many 
feet  ?     How  many  dekameters  long  is  your  schoolroom  ? 

Ten  dekameters  make  one  hektometer.  Hekto  is  a  Greek 
prefix  meaning  hundred.  How  many  meters  in  one  hekto- 
meter ?     How  many  feet  long  is  a  hektometer  ? 

Ten  hektometers  make  one  kilometer.  Kilo  is  a  Greek  prefix 
meaning  thousand.  How  many  meters  in  one  kilometer  ?  How 
many  feet  ?     What  part  of  a  mile  ? 

Ten  kilometers  make  one  myriameter.  Myria  is  a  Greek 
prefix  meaning  ten  thousand.  How  many  meters  in  one  myria- 
meter?    How  many  feet  ?     How  many  miles  ? 


I 


THE  METRIC   SYSTEM  249 

These  statements  may  be  combined  in  the  following  : 

Table  of  Linear  Measure 
10  millimeters  (mm.)  =  1  centimeter  (cm.) 
10  centimeters               =  1  decimeter  (dm.) 
10  decimeters                =  1  meter  (m.) 
10  meters                        =  1  dekameter  (Dm.) 
10  dekameters               =  1  hektometer  (Hm.) 
10  hektometers              =  1  kilometer  (Km.) 
10  kilometers                 =  1  myriameter  (Mm.) 
1 1 1 1 1 1 i 1 i""i"'i| 

One  Decimeter 


l"'l|  IIM1 

One  Centimeter 

n 

One  Millimeter 

435.    Oral 

Read  the  following  expressions  as  meters;  thus,  seventy 
thousand  meters^  fifteen  thousand  meters^  six  hundred  meters^ 
eighty  meters^  one  hundred  fifty-two  thousandths  meters : 

1.  7  Mm.  9.  34  m.  17.  5  Dm.  25.  6  dm. 

2.  15  Km.  10.  7  cm.  18.  61  Km.  26.  47  mm. 

3.  6  Hm.  11.  69  Hm.  19.  384  mm.  27.  523  Km. 

4.  8  Dm.  12.  46  Dm.  20.  7856  m.  28.  368  Dm. 

5.  483  m.  13.  931  Km.  21.  35  cm.  29.  42  Mm. 

6.  8  dm.  14.  26  Hm.  22.  421  mm.  30.  58  Km. 

7.  67  cm.  15.  3  dm.  23.  89  Dm.  31.  284  Dm. 

8.  152  mm.  16.  341  mm.  24.  58  Hm.  32.  700  cm. 

Practice  reading  such  expressions  as  the  above  in  meters,  until  you  cau 
think  in  meters. 


250  GRAMMAR  SCHOOL  ARITHMETIC 


436. 

1  myriameter 
10  kilometers 
100  hektometers 
1000  dekaraeters 
10000  meters 
100000  decimeters 
1000000  centimeters 
10000000  millimeters. 


leducl 

don 

1  millimeter 

O 

.1  centimeter 

.01  decimeter 

fti 

.001  meter 

.0001  dekameter 

3 

.00001  hektometer 

0 

.000001  kilometer 

.0000001  myriameter, 


437.  The  following  series  of  numbers  read  from  the  top 
is  reduction  descending ;  read  from  the  bottom  is  reduction 
ascending.     All  metric  numbers  may  be  reduced  in  this  way. 

7.5689132  Mm.  = 

75.689132  Km.  = 

756.89132  Hra.  = 

7568.9132  Dm.  = 

75689.132  m.     = 

756891.32  dm.   = 

7568913.2  cm.    = 

75689132   mm. 

a  a  a  a    .  i  a  a* 
j^wwoa-sia 

Each  of  these  numbers  may  be  read  thus  :7568913  2. 

438.  Oral  and  Written 

1.  How  may  a  metric  number  be  reduced  to  higher  denomi- 
nations?    To  lower  denominations? 

2.  Reduce  12,345,678  mm.  to  cm.;  todm. ;  to  m.;  to  Dm.; 
to  Hm.:  to  Km.;  to  Mm. 


THE  METRIC  SYSTEM  251 

3.  Reduce  9.6538714  Mm.  to  Km.;   to  Hm.;   to  Dm.;   to 
m.;  to  dm.;  to  cm.;  to  mm. 

4.  Reduce  7  Mm.  to  lower  denominations. 

5.  Reduce  7  mm.  to  higher  denominations. 

6.  Reduce  6307.1  m.  to  Km.;  to  cm. 

7.  Reduce  81  meters  to  inches. 

8.  Write  as  meters  2  Mm.;   7  Km.;   6  Hm.;   8  Dm.;  5  m.; 
3  dm.;  2  cm.;  9  mm.     Write  them  all  as  one  number. 

9.  Reduce  1  Mm.  to  feet. 

10.  Write  7  Mm.  and  6   mm.  in  one    number,   as   meters. 
Reduce  it  to  higher  denominations  ;  to  lower  denominations. 

11.  Reduce  .075  Km.  to  cm. 

12.  Reduce  8  Dm.  and  6  m.  to  Mm. ;  to  mm. 

13.  Write  75  Km.  and  62  dm.  in  one  number  as  meters ;  as 
cm.;  as  Mm. 

14.  State  the  value  of  each  figure  in  30769.543  m. 

15.  A  ship  sails  100    Mm.  in  one  day.     How  many  miles 
does  it  sail  ? 

16.  Give  the  table  of  Metric  Linear  Measure. 

17.  Name  the  standard  unit. 

18.  How  is  it  determined  ? 

19.  What  is  the  scale  of  the  Metric  system  ? 

20.  a.  What  is  the  distance  in  meters  between  two  places  if 
they  are  94,488  feet  apart  ? 

h.    What  is  the  distance  in  kilometers? 

21.  A  boy  in  Paris  walked  12  Km.  in  one  day.      How  many 
miles  did  he  walk  ? 

22.  A  train  in  Europe  ran  393.7  mi.  in  10  hr.     That  was  an 
average  of  how  many  kilometers  per  hour  ? 


252  GRAMMAR   SCHOOL   ARITHMETIC 

SURFACE  MEASURE 

439.  Draw  a  square  whose  side  is  one  meter.  How  many 
square  meters  does  it  contain  ?  It  is  how  many  decimeters  on 
a  side  ?  How  many  square  decimeters  does  it  contain  ?  How 
many  square  decimeters  make  one  square  meter  ? 


One 
sq.  cm. 


\ 


^^. 


One  Square  Decimeter 


How  many  centimeters  long  and  wide  is  a  square  decimeter  ? 
How  many  square  centimeters  in  one  square  decimeter  ?  Find 
how  many  square  millimeters  in  1  sq.  centimeter. 


SURFACE  MEASURE  253 

How  many  sq.  meters  =  1  sq.  dekameter  ? 

How  many  sq.  dekameters     =  1  sq.  hektometer? 
How  many  sq.  hektometers   =  1  sq.  kilometer  ? 

The  answers  to  the  above  questions  form  the  following  table, 
which  is  used  for  all  ordinary  surface  measurements ; 

Table  of  Surface  Measure 

100  sq.  millimeters     =  1  sq.  centimeter  (sq.  cm.) 
100  sq.  centimeters    =  1  sq.  decimeter  (sq.  dm.) 
100  sq.  decimeters     =  1  sq.  meter  (sq.  m.) 
100  sq.  meters  =  1  sq.  dekameter  (sq.  Dm.) 

100  sq.  dekameters    =  1  sq.  hektometer  (sq.  Hm.) 
100  sq.  hektometers  =  1  sq.  kilometer  (sq.  Km.) 

440.    Oral 

1.  Which  denomination  of  our  measure  is  nearest  like  the 
square  meter  ? 

2.  The  square  dekameter  is  equivalent  to  about  how  many 
square  rods  ? 

3.  How  many  square  centimeters  in  one  square  meter  ? 

4.  How  far  to  the  right  must  the  decimal  point  be  moved  to 
reduce  square  meters  to  square  decimeters  ? 

5.  How  many  places  to  the  left  must  the  decimal  point  be 
moved  to  reduce  square  meters  to  square  dekameters  ? 

6.  To  reduce  sq.  mm.  to  sq.  cm.  ? 

7.  To  reduce  sq.  mm.  to  sq.  dm.  ? 

8.  How  many  places  to  the  left  must  the  decimal  point  be 
moved  to  reduce  square  meters  to  square  kilometers  ? 


254  GRAMMAR  SCHOOL   ARITHMETIC 

441.    Written 

1.  Reduce  74.5  square  meters  to  square  centimeters. 

2.  Reduce 

a.  2408  sq.  mm.  to  square  meters. 

b.  .0753  sq.  m.  to  square  millimeters. 

c.  984,769,302  square  meters  to  square  kilometers. 

d.  24.8  sq.  dm.  to  square  centimeters. 

e.  48  sq.  Km.  73  sq.  Dm.  to  square  meters. 

3.  A  table  top  2.5  m.  long  and  95  cm.  wide  contains  how 
many  square  meters  ? 

4.  How  many  square  meters  are  there  in  a  floor  8  m.  long 
and  3  m.  75  cm.  wide  ? 

5.  Find  the  cost  of  painting  the  four  walls  of  a  room  4.5  m. 
long,  3.2  m.  wide,  and  32  dm.  high,  at  1.4  francs  per  square 
meter. 

6.  -Find  in  square  meters  the  entire  surface  of  a  cube  whose 
edge  is  125  cm. 

7.  How  many  square  meters  of  carpet  will  cover  a  floor  896 
cm.  long  and  50  dm.  wide? 

8.  A  city  lot  is  45  m.  long  and  contains  922.50  sq.  m.  of 
land.     Find  its  width  in  centimeters. 

9.  At  30^  per  square  meter,  what  will  it  cost  to  plaster  the 
sides  and  ceiling  of  a  room  5.5  m.  long,  4  m.  wide,  and  3  m. 
95  cm.  high? 

10.  How  many  square  decimeters  of  writing  surface  are  there 
in  a  tablet  containing  90  sheets  of  paper,  each  2  dm.  long  and 
16  cm.  wide  ? 

11.  Find  the  area  of  your  schoolroom  floor  in  square  meters. 

12.  Find  in  square  decimeters  the  area  of  a  square  whose 
edge  is  393.7  inches. 


LAND  MEASURE  255 

LAND   MEASURE 

442.  The  are  (pronounced  air}  and  hectare  are  the  principal 
units  of  land  measure. 

The  are  is  equal  to  one  square  dekameter,  and  the  hectare  is 
equal  to  one  hundred  ares, 

443.  Oral 

1.  An  are  is  how  many  meters  long  ?     Wide  ? 

2.  How  many  square  meters  does  the  are  contain  ? 

3.  An  are  is  how  many  inches  long  ?     Feet  ? 

4.  The  are  is  about  how  many  rods  long  ? 

5.  About  how  many  square  rods  does  it  contain  ? 

6.  About  how  many  ares  equal  one  acre  ? 

7.  How  many  ares  does  a  piece  of  land  as  large  as  the  floor 
of  your  schoolroom  contain  ? 

8.  Name  all  the  surfaces  you  can  think  of  that  contain  about 
one  are. 

444.  Written 

1.  a.    A  field  134  m.  long  and  7  Dm.  wide  contains  how 
many  square  meters  of  land  ? 

h.  How  many  ares  ? 

c.  How  many  hectares  ? 

d.  How  many  square  dekameters  ? 

e.  How  many  square  hektometers  ? 
/.  How  many  square  centimeters  ? 

2.  a.    How  many  square  centimeters  in  an  oblong  643  cm. 
long  and  2.5  m.  wide  ? 

h.    How  many  square  millimeters  ? 
c.    How  many  square  kilometers  ? 

3.  One  hectare  is  equal  to  how  many  acres  ? 


256  GRAMMAR  SCHOOL  ARITHMETIC 

VOLUME  MEASURE 

445.  A  cube  whose  edge  is  one  meter  long  contains  how 
many  cubic  meters  ?  It  is  how  many  decimeters  long  ?  Wide  ? 
High  ?  How  many  cubic  decimeters  does  it  contain  ?  How 
many  cubic  decimeters  equal  one  cubic  meter  ? 

A  cube  whose  edge  is  one  decimeter  contains  how  many 
cubic  decimeters  ?  It  is  how  many  centimeters  long  ?  Wide  ? 
High  ?  How  many  cubic  centimeters  does  it  contain  ?  How 
many  cubic  centimeters  equal  one  cubic  decimeter  ? 

A  cube  whose  edge  is  one  centimeter  contains  how  many 
cubic  centimeters  ?  It  is  how  many  millimeters  long  ?  Wide  ? 
High  ?  It  contains  how  many  cubic  millimeters  ?  How  many 
cubic  millimeters  equal  one  cubic  centimeter  ? 

From  the  answers  to  the  above  questions  make  the  following  : 

Table  of  Volume  Measure 
1000  cu.  millimeters  (cu.mm.)  =  1  cu.  centimeter  (cu.  cm.) 
1000  cu.  centimeters  =  1  cu.  decimeter  (cu.  dm.) 

1000  cu.  decimeters  =  1  cu.  meter  (cu.  m.) 

446.  The  unit  chiefly  used  in  measuring  wood  and  stone  is 
the  stere  (pronounced  stair')^  which  is  a  cube  whose  edge  is  one 
meter.  What  denomination  in  the  English  volume  measure  is 
most  nearly  like  the  stere  ?  How  many  cubic  meters  does  the 
stere  contain  ? 

447.  Oral  and  Written 

1.  How  may  cubic  millimeters  be  reduced  to  cubic  centi- 
meters ?     To  cubic  decimeters  ?     To  cubic  meters  ? 

2.  How  many  places  to  the  right  must  the  decimal  point 
be  moved  to  reduce  cubic  meters  to  cubic  millimeters  ? 


CAPACITY  MEASURE  257 

3.  Reduce  7  cubic  meters  to  cubic  millimeters. 

4.  Reduce  5  cubic  millimeters  to  cubic  meters. 

5.  How  many  steres  in  one  cubic  meter? 

6.  A  pile  of  wood  is  30  dm.  long,  3  m.  wide,  and  18  dm. 
high,     a.    How  many  cubic  meters  does  it  contain? 

h.    How  many  steres? 

c.    How  many  cubic  millimeters? 

7.  a.    How  many  cubic  centimeters  of  air  in  an  empty  box 
2  m.  by  12  dm.  by  75  cm.  ? 

h.    How  many  cubic  decimeters? 

8.  How  many  steres  of  stone  in  a  wall  30  m.   long,  5  dm. 
thick,  and  250  cm.  high? 

CAPACITY  MEASURE 

448.  The  metric  capacity  measure  takes  the  place  of  both  the 
liquid  and  the  dry  measure  of  the  English  system. 

The  standard  unit  of  capacity  measure  is  the  liter  (pronounced 
leeter)^  which  is  a  cube  whose  edge  is  one  decimeter. 

449.  Oral  and  Written 

1.  The  liter  is  what  part  of  a  meter  wide?     High?     Long? 

2.  What  part  of  a  cubic  meter  does  it  contain? 

3.  About   how  many  inches  wide  is  it?      High?      Long? 
About  how  many  cubic  inches  does  it  contain? 

4.  Show   with   your   hands    how   wide,   high,    and   long    a 
liter  is. 

5.  What  denomination  of  English  dry  measure  corresponds 
most  nearly  to  the  liter? 

6.  Make  a  full-sized  picture  of  a  liter. 

7.  What  object  the  size  of  a  liter  do  you  know? 


258  GRAMMAR   SCHOOL   ARITHMETIC 

Table  of  Capacity  Measure 

450.  The  table  of  capacity  measure  is  formed  similarly  to  the 
other  metric  tables,  and  is  as  follows : 

10  milliliters  (ml.)  ==  1  centiliter  (cl.) 

10  centiliters  =  1  deciliter  (dl.) 

10  deciliters  =  1  liter  (1.) 

10  liters  =  1  dekaliter  (Dl.) 

10  dekaliters  =  1  hektoliter  (HI.) 

10  hektoliters  =  1  kiloliter  (Kl.) 

10  kiloliters     .  =1  myrialiter  (Ml.) 

451.  Oral  and  Written 

1.  How  many  liters  in  1  myrialiter?     In  1  milliliter? 

2.  How  many  milliliters  in  1  myrialiter?  i 

3.  Reduce  12,345,678  ml.  to  higher  denominations. 

4.  Read  the  number  in  example  3,  giving  each  figure  the 
name  of  the  denomination  it  represents. 

5.  Reduce  154.67  cl.  to  kiloliters. 

6.  Reduce  .012346  Ml.  to  deciliters. 

7.  How  many  liters  equal  one  cubic  meter? 

8.  A  bin  is  2.5  m.  wide,  6.4  m.  long,  and  17  dm.  deep. 
How  many  liters  of  oats  will  it  hold?  How  many  hektoliters? 
How  many  kiloliters  ? 

9.  A  tank  is  3  m.  long  and  3  m.  wide.  How  many  deci- 
meters deep  must  it  be  to  hold  50  HI.  of  water  ? 

10.    A  stone  whose  volume  is  1  stere,  if  dropped  into  a  pond, 
would  displace  how  many  liters  of  water? 


MEASURES   OF   WEIGHT  259 

MEASURES  OF   WEIGHT 

452.  The  gram  is  the  unit  of  weight.  It  is  equal  to  the 
weight  of  a  cubic  centimeter  of  distilled  water  at  its  greatest 
density.     One  gram  equals  15.432  grains. 

Table  of  Weight 
10  milligrams  (mg.)  =  1  centigram  (eg.) 
10  centigrams  =1  decigram  (dg.) 

10  decigrams  =1  gram  (g.) 

10  grams  •       =1  dekagram  (Dg.) 

10  dekagrams  =  1  hektogram  (Hg.) 

10  hektograms  =  1  kilogram  (Kg.) 

10  kilograms  =  1  myriagram  (Mg.) 

10  myriagrams  =  1  quintal    (Q.) 

10  quintals  =  1  tonneau,  1  ^T  ^ 

or  metric  ton  J 

453.  Oral  and  Written 

1.  How  many  grams  in  1  metric  ton? 

2.  How  many  myriagrams  in  1  metric  ton? 

3.  Reduce  1  mg.  to  metric  tons. 

4.  Reduce  1  T.  to  milligrams. 

5.  Reduce  9,876,543,215  mg.  to  higher  denominations. 

6.  Read  the  number  in  example  5,  giving  each  figure  the 
name  of  the  denomination  it  represents. 

7.  Recite  the  table  of  weight. 

8.  Spell  the  name  of  each  denomination. 

9.  Reduce  7.42  quintals  to  centigrams. 

10.  Reduce  543  mg.  to  myriagrams. 

11.  How  many  grains  in  1  Kg.  ?  . 


260  GRAMMAR  SCHOOL  ARITHMETIC 

12.  One  pound  Avoirdupois  contains  7000  gr.     How  many 
pounds  are  equivalent  to  one  kilogram  ? 

13.  Mr.  Smith  weighs  100  Kg.     How  many  pounds  does  he 
weigh  ? 

14.  How  many  grams  does  a  cubic  meter  of  distilled  water 
weigh  ? 

15.  Would  a  cubic  meter  of  any  other  substance  weigh  the 
same  as  a  cubic  meter  of  distilled  water  ?     State  your  reason. 

16.  How  many  kilograms  of  water  will  a  tank  4  m.  x  3  m. 
X  12  dm.  hold? 

REVIEW   QUESTIONS 
454.   1.   How  many  tables  are  there  in  the  Metric  System  ? 

2.  Name  the  standard  units  in  the  order  in  which  they  have 
been  given.  Repeat  them  until  you  can  say  them  as  rapidly 
as  you  can  talk. 

3.  Name  the  prefixes  in  the  same  way. 

4.  Name  and  describe  the  unit  of  capacity  measure ;  of 
weight ;  of  length ;  of  volume ;  of  surface. 

5.  Repeat  the  tables. 

6.  The  stere  is  the  unit  of  what  measure  ?  The  meter  ? 
The  are  ?     The  gram  ?     The  liter  ? 

7.  How  can  metric  numbers  be  reduced  to  higher  denomi- 
nations ?    To  lower  ? 

8.  How  many  things  are  to  be  committed  to  memory  in  the 
Metric  System  ? 

9.  What  is  39.37?  15.432?  10?  These  are  the  only 
numbers  that  need  be  remembered. 


DUTIES  261 


DUTIES 


455.  Under  the  head  of  taxes^  page  220,  we  discussed  the 
methods  of  raising  money  for  the  support  of  city,  village, 
township,  county,  and  state  governments.  These  are  chiefly 
methods  of  direct  taxation;  that  is,  the  taxes  are  paid  directly 
by  all  owners  of  property  and  are  apportioned  according  to  the 
assessed  valuation  of  the  property. 

The  expenses  of  the  national  government  are  great.  Vast 
sums  of  money  are  required  for  the  support  of  the  army  and 
navy,  payment  of  pensions  to  veteran  soldiers  and  sailors,  pay- 
ment of  the  salaries  of  the  President,  Vice-President,  senators, 
representatives,  and  other  officers  and  employees  of  the  govern- 
ment, building  of  post  offices  and  other  public  buildings, 
improvement  of  rivers  and  harbors,  keeping  of  lighthouses  and 
life-saving  stations,  and  for  many  other  purposes.  Name  other 
expenses  of  the  national  government. 

The  expenses  of  the  post  office  department  are  largely  paid 
by  the  sale  of  postage  stamps.  This  is  a  tax  upon  the  persons 
buying  the  stamps,  but  they  receive  an  immediate  and  direct 
return  by  having  their  mail  carried.  There  are  two  other 
means  by  which  most  of  the  money  for  government  use  is  ob- 
tained; namely, 

a.   By  internal  revenue  taxes. 

h.    By  duties  or  customs. 

456.  Internal  revenue  taxes  are  taxes  levied  on  certain  arti- 
cles made  in  this  country,  chiefly  spirits  and  tobacco  products. 
It  is  unlawful  to  sell  these  articles  before  .the  internal  revenue 
tax  upon  them  has  been  paid,  and  persons  who  break  the  law 
may  be  punished  by  fine  or  imprisonment. 

457.  Duties  or  customs  are  taxes  levied  on  certain  articles 
imported  into  the  country  from  foreign  lands. 


262  GRAMMAR  SCHOOL   ARITHMETIC 

Most  articles,  other  than  those  subject  to  internal  revenue 
taxes,  may  be  produced  or  manufactured  in  this  country/  with 
entire  freedom  and  without  taxation ;  but  there  are  many- 
things,  both  manufactured  articles  and  "raw  materials,"  that 
cannot  be  brought  into  the  country  without  having  taxes  levied 
upon  them  and  collected  by  the  government.  These  taxes, 
called  duties  or  customs,  are  collected  at  custom  houses,  located 
at  cities  and  towns  called  ports  of  entry.  The  ports  of  entry 
are  situated  not  only  along  the  seacoast  and  other  boundaries 
of  the  country,  but  also  along  the  great  river  and  railroad 
routes.     Can  you  name  some  cities  that  are  ports  of  entry  ? 

458.  Articles  on  which  duty  must  he  paid  are  called  dutiable 
articles.  It  is  unlawful  for  dutiable  articles  to  be  brought  into 
the  country  at  any  other  place  than  a  port  of  entry. 

459.  A  list  of  dutiable  articles  and  the  rates  of  duty  to  he  paid 
upon  them  is  called  a  tariff.  The  tariff  of  the  United  States  is 
fixed  by  Congress. 

The  importer  of  foreign  goods  must  pay  the  duty  on  goods  which  he 
imports.  Therefore,  when  he  sells  the  goods,  he  must  ask  a  price  sufficient 
to  cover  the  cost,  the  duty  paid,  and  his  profit ;  so  that  the  person  who 
finally  buys  the  goods  for  his  own  use  really  pays  the  duty  upon  them.  The 
duty  or  custom  is  therefore  said  to  be  an  indirect  tax  upon  the  purchaser  or 
consumer. 

460.  Duty  computed  at  a  certain  per  cent  of  the  cost  of  the 
goods  in  the  country  from  which  they  mere  shipped  is  ad  valorem 
duty;  e.g.  the  duty  on  f  10,000  worth  of  laces  at  60%  ad 
valorem  is  $6000. 

461.  Duty  computed  according  to  the  quantity  of  goods  im- 
ported is  specific  duty;  e.g.  the  duty  on  10,000  lb.  of  currants 
at  2  cents  per  pound  is  f  200. 


DUTIES 


263 


Some  articles  are  subject  to  both  an  ad  valorem  and  a  specific 
duty;  e.g.  the  duty  on  cotton  wicking  is  15%  ad  valorem 
and  10  cents  per  pound. 

462.  Tare  is  an  allowance  made  for  the  weight  of  boxes  or 
cases  in  which  goods  are  packed  for  shipment. 

463.  Leakage  and  breakage  are  allowances  for  loss  of  liquids 
shipped  in  barrels,  casks,  and  bottles. 

464.  In  computing  ad  valorem  duty,  take  the  net  foreign  in- 
voice valuation  (value  of  the  goods  in  the  money  of  the  coun- 
try from  which  they  were  shipped,  less  all  discounts),  find  its 
exchange  value  in  United  States  money,  and  find  the  required 
per  cent  of  that  sum.  If  the  valuation  contains  a  fraction  of 
a  dollar  equal  to,  or  greater  than,  fifty  cents,  call  it  another 
dollar ;  if  less  than  fifty  cents,  omit  it ;  e.g.  a  case  of  cotton 
laces  invoiced  at  £  100,  less  4  %,  is  valued  at  £  96,  or  $467,  and 
the  duty  is  60%  of  $467,  or  $280.20. 

In  changing  the  foreign  invoice  valuations  to  dollars,  use  the 
following  rates,  which  represent  the  intrinsic  par  or  real  com- 
parative values  of  the  various  denominations,  as  adopted  by  the 
United  States  Treasury  Department. 


Country 

Monetary  Unit 

Value  in  U.  S. 
Dollars 

Great  Britain 

Germany 

France          "j 

Switzerland  > 

Belgium        j 

Italy 

Austria 

Pound 
Mark 

Franc 

Lira 
Crown 

$4,866 
$.238 

$.193 

$.193 
$.203 

264 


GRAMMAR  SCHOOL  ARITHMETIC 


465.    Oral 

Find  the  duties  on  the  following  invoices 
Articles 

1.  500  lb.  of  figs 

2.  $200  worth  of  cotton-seed  meal 

3.  800  lb.  macaroni 

4.  $2000  worth  of  mandolins 

5.  2  T.  of  mutton 

6.  50,000  white  pine  shingles 

7.  5000  bu.  of  apples 

8.  4855  lb.  lemons 

9.  2500  pineapples 

10.  i  200  worth  of  straw  hats 

11.  $480  worth  of  artists'  proof  etchings 

12.  15  cwt.  of  Italian  chestnuts 

13.  One  ton  of  hydraulic  cement 

14.  10  horses,  valued  at  $300  apiece 

15.  $200  worth  of  silk  gloves 

16.  50  bu.  of  flaxseed 

17.  2  T.  of  maple  sugar 

18.  $150  worth  of  rubber  balls 

19.  $2100  worth  of  steel  plows 

20.  5  T.  of  car  tires 

21.  800  lb.  of  frozen  salt-water  fish 

22.  $1000  worth  of  sawed  mahogany 

23.  600  bottles  of  Apollinaris  water 

24.  5  T.  of  scoured  wool 

25.  500  knives  invoiced  at  40^  each 


Rate  of  Duty 

2^  per  pound. 

20  per  cent. 

IJ^  per  pound. 

45  per  cent. 

2^  per  pound. 

30^  per  1000. 

25^  per  bushel. 

1^  per  pound. 

$7  per  1000 

35  per  cent. 

25  per  cent. 

1^  per  pound. 

8^  per  100  pounds. 

25  per  cent. 

60  per  cent. 

25  ^  per  bushel. 

4^  per  pound. 

30  per  cent. 

20  per  cent. 

IJ^  per  pound. 

I  ^  per  pound. 

15  per  cent. 

30^  per  dozen  bottles. 

33/  per  pound. 

5/  each  and  40%. 


DUTIES  265 

466.     Written 

In  examples  1-15  compute  the  duties  in  dollars  : 

1.  On  $1275  worth  of  chisels  at  45%. 

2.  On  13842  worth  of  fur  rugs  at  35  %. 

3.  On  500  bbl.  of  rye  flour,  each  containing  196  lb.,  at  J^ 
per  pound. 

4.  On  $8374  worth  of  wool  garments  weighing  1047  lb.,  at 
44^  per  pound  and  60  %  ad  valorem. 

5.  35%  on  1893  yd.  of  gingham,  invoiced  at  13^  per  yard. 

6.  2J^  per  square  yard  on  648  sq.  yd.  of  unbleached  cotton 
cloth. 

7.  60  %  ad  valorem  and  44  ^  per  pound  on  8  cases  of  wool 
stockings,  average  weight  per  case  272  lb.,  invoiced  at  $2685. 

8.  25  %  ad  valorem  and  $  3  apiece  on  25  Swiss  watches, 
valued  at  165  apiece. 

9.  On  350  lb.  of  cologne  water,  invoiced  at  40/  per  pound, 
the  rate  being  45%  ad  valorem  and  60/  per  pound. 

10.  Five  tons  of  corrugated  iron  plates  at  lyo^  P®^  pound. 

11.  20%  ad  valorem  and  60/  per  square  yard  on  500  yd.  of 
inlaid  linoleum,  6  ft.  wide,  invoiced  at  60/  per  square  yard. 

12.  On  504  dozen  boxes  of  friction  matches  at  8  /  per  gross 
of  boxes. 

13.  60  /  per  square  yard  and  40  %  ad  valorem  on  525  yd.  of 
Wilton  carpet,  27  in.  wide,  invoiced  at  80/  per  yard. 

14.  4/  per  pound  and  15  %  on  1500  lb.  of  candy,  invoiced 
at  15/  per  pound. 

15.  35%  on  a  shipment  of  fur  coats  from  Kraft  and  Levin, 
Berlin,  invoiced  at  3192  marks,  less  4%. 


266  GRAMMAR   SCHOOL   ARITHMETIC 

16.  Henry  Johnson  of  Denver  purchased  from  the  Broadway 
Damask  Co.  of  Belfast,  Ireland,  1168  sq.  yd.  of  linen  damask, 
invoiced  at  X88,  less  3%. 

a.    Find  the  net  invoice  price  in  dollars. 

h.    Compute  the  duty  at  30  %  and  6^  per  square  yard. 

17.  Williams  &  Co.  of  Cleveland  bought  of  Moritz  Pach 
of  Berlin,  15  wool  jackets,  weighing  20  Kg.,  invoiced  at  873 
marks;  8  wool  coats  weighing  12  Kg.,  for  798  marks;  and  9 
silk  coats  for  1068  marks.  The  purchasers  were  allowed  a 
4  %  trade  discount  on  the  entire  in  voice.  They  paid  a  duty  of 
44  ^  per  pound  and  60  %  ad  valorem  on  the  wool  garments,  and 
60  %  ad  valorem,  only,  on  the  silk  garments. 

a.    Find  in  marks  the  net  price  of  the  entire  invoice. 
h.    Find  in  dollars  the  net  price  of  the  entire  invoice. 

c.  Find  in  pounds  the  weight  of  the  wool  garments  (to  tenths). 

d.  What  was  the  amount  of  duty  paid  ? 

18.  Mr.  M.  J.  McCarthy  purchased  of.  E.  J.  Weinfurter, 
Vienna,  Austria,  410  Kg.  of  candle  wicking,  invoiced  at  2460 
crowns,  less  5  %  trade  discount. 

a.    What  was  the  net  invoice  price  in  crowns  ? 

h.    What  was  the  net  invoice  price  in  United  States  money  ? 

c.  What  was  the  duty,  computed  at  10/  per  pound  and 
15  %  ad  valorem  ? 

d.  What  was  the  total  cost  of  the  goods,  including  the  net 
invoice  price,  the  duty,  15  crowns  for  cases  and  packing,  and 
12.40  crowns  for  consular  certificates  ? 

19.  Leighton  and  McArthur  of  Rochester  bought  from  San- 
derson Brothers  and  Newbould,  of  Sheffield,  England,  12,518  lb. 
of  steel  ingots  invoiced  atX811  9s.  Id.  What  was  the  entire 
cost  in  United  States  money,  including  a  duty  of  4^^  cents  per 
pound,  freight  ^6  5s.  2c?.,  commissions  18s.  9c?.,  consular  fees 
10s.  4c?.  and  insurance  X 1  6s.  ? 


EQUATIONS  267 

20.  Fanclier  and  Dunham  of  Providence  purchased  of  the 
Compagnie  de  Vichy  of  Lyons,  France,  120  cases  of  mineral 
water,  each  containing  50  quart  bottles,  invoiced  at  35  francs 
per  case,  and  5  cases,  each  containing  100  pint  bottles,  invoiced 
at  45  francs  per  case.  Find  in  United  States  money  the  entire 
cost,  including  a  duty  of  20)^  per  dozen  pint  bottles  and  30/  per 
dozen  quart  bottles. 

21.  What  is  the  duty  at  60  %  on  a  case  of  cotton  laces  con- 
taining 1090  pieces  purchased  from  the  Thomas  Adams  Co., 
Limited,  of  Nottingham,  England,  invoiced  at  8Jc?.  per  piece 
with  trade  discounts  of  20  %  and  5  %  ? 

22.  When  the  duty  on  32,500  pine  shingles  amounts  to 
19.75,  what  is  the  duty  per  1000  ? 

23.  When  a  45%  duty  on  an  invoice  of  goods  from  France 
amounts  to  $260.55,  what  is  the  invoice  price,  in  French  money? 

24.  A  shipment  of  goods  from  Austria  was  invoiced  at  3500 
crowns.     What  was  the  ad  valorem  duty  at  15  %  ? 

25.  Find  the  amount  of  a  6  %  duty  on  goods  invoiced  at  &  200. 

EQUATIONS 

467.  An  expression  of  the  equality  of  two  numbers  or  quantities 
is  an  equation;  e.g. 

$40  =  140;        32oz.  =2  1b. ;        |20x2  =  |40; 
8  cents  -f-  2  =  4  cents ;        ^  1  5s.  =  25«. 

468.  The  part  of  an  equation  at  the  left  of  the  sign  of  equality 
is  the  first  member  of  the  equation. 

469.  The  part  of  an  equation  at  the  right  of  the  sign  of  equality 
is  the  second  member  of  the  equation. 

Name  the  first  member  of  each  of  the  equations  in  section 
467  ;  the  second  member. 


268 


GRAMMAR  SCHOOL  ARITHMETIC 


Fig.  1 


Fig.  4 


Fig.  7 


Fig.  2 


Fig.  5 


Fig.  8 


Fig.  3 


Fig.  6 


Jsib.-P^  r'eibT 

Fig.  9 


470.    Oral 

1.  Which  of  the  above  figures  represent  equations  ? 

2.  Why  do  the  scales  balance  in  Fig.  1  ? 

3.  Why  do  they  not  balance  in  Fig.  2  ? 

4.  What  must  be  done  with  Fig.  2  to  obtain  the  balance 
shown  in  Fig.  3  ? 

5.  What  must  be  done  with  Fig.  4  to  obtain  the  balance 
shown  in  Fig.  5  ? 

6.  What  must  be  done  with  Fig.   6  to  obtain  the  balance 
shown  in  Fig.  7  ? 


EQUATIONS  269 

7.  What  must  be  done  with  Fig.  8  to  obtain  the  balance 
shown  in  Fig.  9  ? 

8.  Write  an  equation  expressed  in  dollars.     Add  $5  to 
each  member.     Is  it  still  an  equation?     Why? 

9.  How  may  we  make  a  true  equation  from  17  =  14? 

10.  How  may  we  make  a  true  equation  from  21  =  7  ? 

11.  How  may  a  true  equation  be  made  from  15  gal. -5- 3 
=  60qt.? 

12.  Complete  the  following  equations  : 

a.  85+ =45.  h.  $99^  11  =  11  X . 

h.   89-3=80  + .  {.  86-46  =  5x— . 

c.  45  =  15  X .  y.  5  ft.  +  8  in.  =  60  in  + . 

d.  17  ft.  =  5  yd. ft.         k.  2  hr.  +  30  min.  =  min. 

e.  2rd.  7  ft.  =  32ft. +  - ,  I.  |  =  ^. 

/.    4wk.  = da.  w.  41= . 

g,   18  +  3  X  6  =  30  + .     n,  x  7  =  60  - 11. 

13.  Make  an  equation.  Add  7  to  the  first  member.  Is  it 
still  an  equation  ?  What  must  be  done  to  the  second  member 
to  restore  the  equality  ? 

14.  Make  an  equation  of  two  sums  of  money.  Add  10  cents 
to  the  first  member.  What  must  be  done  to  the  second  mem- 
ber in  order  to  preserve  the  equality  ? 

15.  Make  an  equation  of  two  numbers  expressing  time. 
Subtract  15  min.  from  the  second  member.  What  must  be 
done  to  the  first  member  to  preserve  the  equality? 

16.  Make  an  equation  of  two  numbers  expressing  surfaces. 
Multiply  both  members  by  10.  How  is  the  equality  of  the  two 
members  of  the  equation  affected? 

17.  Make  an  equation.  Divide  both  members  by  the  same 
number.     How  is  the  equality  of  the  two  members  affected  ? 


270  GRAMMAR  SCHOOL  ARITHMETIC 

471.  Axioms 

1.  If  the  same  or  equal  quantities  are  added  to  equal  quantities^ 
the  sums  are  equal. 

2.  If  the  same  or  equal  quantities  are  subtracted  from  equal 
quantities^  the  remainders  are  equal. 

3.  If  equal  quantities  are  multiplied  by  the  same  or  equal  quan- 
tities^ the  products  are  equal. 

4:.  If  equal  quantities  are  divided  by  the  same  or  equal  quan- 
tities^ the  quotients  are  equal. 

Summary 

We  may  add  the  same  number  or  equal  numbers  to  both  mem- 
bers of  an  equation.,  subtract  the  same  number  or  equal  numbers 
from  both  members  of  an  equation.,  midtiply  both  members  by  the 
same  or  equal  numbers^  or  divide  both  members  by  the  same  or 
equal  numbers  without  destroying  the  equality. 

472.  Many  problems  may  be  solved  more  easily  by  the  use 
of  equations  than  by  the  usual  methods  of  analysis.  In  solving 
problems  by  means  of  equations,  it  is  customary  to  represent 
the  number  which  is  to  be  found.,  called  the  unknown  number,  by 
some  letter,  usually  x^  ?/,  or  z. 

In  expressing  the  equation,  if  x  stands  for  a  certain  number, 
two  times  the  number  is  represented  by  2  a;,  three  times  the 
number  by  3  a;,  ten  times  the  number  by  10  Xy  and  so  on ;  that 
is,  5  X  means  5  times  x.,1  x  means  7  times  a?,  25  x  means  25  times  x., 
.05  a?  means  .05  of  a;,  and  so  on. 

What  is  the  meaning  of  11a;?  15a;?  fa;?  7Ja;?  .15a;? 
2.07a;?  .03a;?  2|a;? 

473.  Finding  the  value  of  the  unknown  number  in  an  equation 
is  called  solving  the  equation. 


EQUATIONS  271 

We  solve  an  equation  by  adding  the  same  or  equal  numbers 
to  both  members,  subtracting  the  same  or  equal  numbers  from 
both  members,  multiplying  both  members  by  the  same  or  equal 
numbers,  or  dividing  both  members  by  the  same  or  equal  num- 
bers, or  by  performing  several  of  these  operations  in  succession. 
In  other  words,  there  are  four  operations  that  we  may  perform 
upon  the  members  of  an  equation  without  destroying  the  equality. 

Examples 

1.  Solve  the  equation,  8  a;  =  24, 
Dividing  both  members  by  8,  ic  =  3.    A.ns. 

2.  Solve  the  equation,  x  +  15  =  45, 
Subtracting  15  from  both  members,  27=  30.      A.n8. 

3.  Solve  the  equation,  4  a;  +  $10  =  |38, 
Subtracting  f  10  from  both  members,  4  a;  =  #  28, 
Dividing  both  members  by  4,  x  =  $7.    Ans. 

4.  Solve  the  equation,  16  x  +  $20  =6  x  +  $S5, 
Subtracting  6  x  from  both  members,    10  a?  +  $  20  =  $35, 
Subtracting  $20  from  both  members,  10  a;  =  $15, 
Dividing  both  members  by  10,  a:  =$1.50.    Ans. 

5.  Solve  the  equation,  ^  a;  —  18  =  2, 
Adding  18  to  both  members,  i  ^  =  2^» 
Multiplying  both  members  by  5,        '  x  =  100.    Ans. 

6.  Solve  the  equation,  82  ic  —  ^  =  40^\, 
Adding  ^  to  both  members,  82  a;  =  41, 
Dividing  both  members  by  82,  x  =  ^.    Ans. 

7.  Solve  the  equation,  1.00\x==  $84.21, 
Dividing  both  members  by  1.00|,  a?  =  $84.    Ans. 


272 


GRAMMAR  SCHOOL  ARITHMETIC 


474.     Written 

Solve  the  following  equations 

1.  5  2^=35. 

2.  7x=lS  +  4:x. 

3.  13a:4-4  =  95. 

4.  18-1  2:  =  74. 

5.  -^^x  =  SQ, 

6.  I  of  f  2^  =  825. 

7.  1.03  a:  =  412. 

8.  .08  a;  =  4.32. 


9.  5.18:^  =  466.2  ydo 

10.  8.5  2;  +  30bu.  =1135ba 

11.  18  2^-2  =  88. 

12.  75a;-f  =  224f 

13.  14  a;  +  i\  =  560^5^. 

14.  12fa:  =  |957. 

15.  45  a;  =72. 

16.  .36  a; +  11.45  =  119.45. 


Problems 
475.    Written 

1.  .16  of  the  cost  of  my  house  was  1 320.     What  did  my 
house  cost? 

Solution 
Let  X  =  cost  of  my  house. 

Then  .16  a;  =  $320. 

Dividing  both  members  by  .16,  x  =  $2000,  cost  of  my  house.   Ans. 

2.  A  pony  and  cart  cost  f  135.     The  pony  cost  four  times  as 
much  as  the  cart.     Find  the  cost  of  each. 

Solution 
Let  »  =  cost  of  the  cart. 

Then  4  a:  =  cost  of  the  pony. 

Adding,  5  x=^  135,  cost  of  both. 

Dividing  both  members  by  5,       x  =  $  27,  cost  of  the  cart. 
Multiplying  by  4,  4  a;  =  $  108,  cost  of  the  por 

3.  The  sum  of  two  numbers  is  199.40.     Their  difference  is 
2.70.     What  are  the  numbers  ? 


\  Ans. 

>ny-J 


EQUATIONS  273 

Solution 
Let  X  =  the  smaller  number. 

Then  x  +  2.70  =  the  larger  number. 

Adding  equals  to  equals,  2  x  +  2.70  =  the  sum  of  the  numbers 

or,  2  a: +  2.70  =  199.40. 
Subtractmg  2.70  from  both  members,  2x  =  196.70. 
Dividing  both  members  by  2,  x  =  98.35,  the  smaller,  ^ 

Adding  2.70  to  both  members,     x  +  2.70  =  101.05,  the  larger.  |^^*' 

4.  The  area  of  a  rectangle  is  5875  square  inches.     The  width 
is  25  inches.     What  is  the  length  ? 

Solution 
Let  X  =  the  length  in  inches. 

Then  25  x  =  5875  (area  =  length  x  breadth). 

Dividing  both  members  by  25,  x  =  235  inches.     Ans, 

5.  A  merchant  gained  35  %  by  selling  cloth  at  f  1.89  per  yard. 
What  was  the  cost  per  yard  ? 

Solution 
Let  X  =  cost  of  1  yard. 

Then  .35  x  =  gain  on  1  yard. 

Adding  equals  to  equals,  1.35  x  =  cost  +  gain,  or  the  selling  price 

or,  1.35  a:  =  $1.89. 
Dividing  both  members  by  1.35,  x=  $  1 .40,  cost  of  1  yard.   A  ns. 

6.  What  principal  on  interest  for  2  mo.  21  da.  at  5%,  will 
yield  17.47  interest? 


Solution 

Let                                X  =  the  required  principal. 

9    ■ 

100   ^m     800 
8 

Then       xx^l    X  ^^  =$7.47. 
100      360 

Therefore,     ^   x=  $7.47. 
800 

Dividing  both  members  by  - — ,  x  =  $664,  principal.    Ans. 


274  GRAMMAR  SCHOOL  ARITHMETIC 

Solve  hy  means  of  equations : 

7.  John  and  Henry  earned  138.40  during  the  summer  vaca- 
tion. Henry  earned  twice  as  much  as  John.  How  much  did 
each  earn  ? 

8.  The  sum  of  two  numbers  is  834T  ;  their  difference  is  1265. 
What  are  the  numbers  ? 

9.  Elsie,  Ruth,  and  Mabel  received  $42  in  prizes,  Elsie 
receiving  f  3  as  often  as  Ruth  f  2  and  Mabel  %\.  What  was  the 
amount  of  each  prize  ? 

10.  A  pole  stands  -^^  in  the  mud,  ^y  in  the  water,  and  the 
remainder,  which  is  32  feet,  in  the  air.     How  long  is  the  pole? 

Hint.  —  Let  x  =  yV  of  the  length  of  the  pole. 

11.  A  tree  bb  ft.  high  was  broken  off  so  that  the  part  broken 
off  was  four  times  as  long  as  the  part  left  standing.  How  long 
was  the  piece  that  was  broken  off  ? 

12.  Three  men,  A,  B,  and  C,  engaged  in  business,  B  furnish- 
ing three  times  as  much  capital  as  A,  and  C  furnishing  twice 
as  much  as  B.  If  they  furnished  18950  in  all,  how  much  did 
each  furnish? 

13.  A  man  is  four  times  as  heavy  as  his  son,  and  the 
difference  of  their  weights  is  63  Kg. 

a.    What  is  the  weight  of  each,  in  kilograms? 
5.    In  pounds  ? 

14.  What  number  increased  by  \  of  itself  equals  192  ? 
Hint.  —  Let  a:  =  }  of  the  number ;  then  1  x  =  the  number. 

15.  What  number  diminished  by  -^j^^^  of  itself  equals  162  ? 

16.  A  man,  having  a  sum  of  money,  earned  five  times  as 
much,  and  spent  one  half  of  what  he  then  had.  He  had  left 
1270.     How  much  had  he  at  first  ? 


EQUATIONS  275 

17.  A  boy,  having  some  money,  earned  twice  as  much  and 
$.48  more,  when  he  had  19.78.     How  much  did  he  earn? 

18.  One  third  of  a  sum  of  money  exceeds  one  fourth  of  the 
sum  by  $17.     What  is  the  sum  ? 

19.  Two  fifths  of  a  number  is  14  less  than  five  ninths  of  the 
number.     Find  the  number. 

20.  2|  times  a  certain  number  is  greater  by  45  than  three 
fourths  of  the  number.     Find  the  number. 

21.  Divide  176  into  four  parts  so  that  the  first  shall  be  four 
times  the  second,  the  third  one  third  of  the  second,  and  the 
fourth  one  half  of  the  first. 

Hint.  —  Let  x  —  the  third  part. 

22.  The  sum  of  three  numbers  is  1658.  The  second  exceeds 
the  first  by  130,  and  the  third  exceeds  the  first  by  79.  Find  the 
three  numbers. 

23.  Three  numbers,  when  added,  amount  to  11.89.  The 
second  exceeds  the  first  by  3.28.  and  the  third  exceeds  the 
second  by  1.37.     Find  them. 

24.  A  farmer  has  apples,  potatoes,  turnips,  and  onions  in  his 
cellar.  The  number  of  bushels  of  apples  is  13  less  than  the 
number  of  bushels  of  potatoes;  the  number  of  bushels  of 
turnips  is  19  less  than  the  number  of  bushels  of  apples,  and 
there  are  3  more  bushels  of  turnips  than  of  onions.  The  entire 
quantity  is  72  bushels.     Find  the  number  of  bushels  of  each. 

25.  In  a  certain  class,  the  number  of  girls  who  received 
honor  marks  was  three  more  than  twice  the  number  of  boys 
who  received  honor  marks.  The  number  of  honor  pupils  was 
18.     How  many  were  girls,  and  how^  many  were  boys  ? 

26.  Seven  times  a  certain  sum  of  money  plus  $18  is  equal  to 
five  times  the  sum  plus  $50.     What  is  the  sum  of  money  ? 


276  GRAMMAR   SCHOOL   ARITHMETIC 

REVIEW  AND  PRACTICE 
476.    Oral 

1.  Name  the  prime  numbers  from  1  to  100. 

2.  How  may  we  know,  without  trial,  that  723,468  will  not 
exactly  divide  398,650,076,341  ? 

3.  There  are  two  decimal  places  in  one  factor,  three  in 
another,  one  in  another,  and  four  in  another.  How  many 
decimal  places  are  there  in  the  product  of  the  four  factors  ? 

4.  If  one  fifth  of  an  acre  of  land  is  worth  120,  what  is  one 
twentieth  of  an  acre  worth  at  the  same  rate  ? 

5.  48  X  25  =  ?    57  X  99  =  ?    560  x  125  =  ? 

6.  61-v-25  =  ?     33 -V- 125  =  ?     17^.331  =  ? 

7.  360x.l6|  =  ?     39-^.25  =  ?     150-^.2==? 

8.  63  X  33J  =  ?     99  X  66f  =  ?     42  x  .14f  =  ? 

9.  50  +  5x2  =  ?     88-8-^4  =  ?     7x8  +  16-^4=? 

10.  20  %  of  33 J  %  =  what  common  fraction  ? 

11.  Two  successive  trade  discounts  of  10  %  are  the  same  as 
what  single  discount  ? 

12.  Test  each  of  the  following  numbers  for  divisibility  by  2, 
3,  4,  5,  6,  8,  and  9 : 

a.    2364  h,   486,728        e.    72,056,391       d,    91,307,865 

e.    42,836,076  /.    90,010,332  g,    8,705,637,411 

13.  If  a  man  earns  f  99  in  17  days,  how  much  will  he  earn  in 
51  days  at  the  same  rate  ? 

14.  What   is   the   least  number   that   will   exactly  contain 
2,  3,  and  4  ? 

15.  What  is  the  greatest  number  that  will  exactly  divide  60, 
96,  and  132  ? 

16.  What  is  the  cost  of  7000  shingles  at  f  5.50  per  M  ? 


REVIEW  AND  PRACTICE  277 

17.  What  is  the  cost  of  1500  lb.  of  mixed  feed  at  f  1.80 
per  cwt.  ? 

18.  Two  long  tons  contain  how  many  more  pounds  than  two 
short  tons  ? 

19.  What  is  the  length  of  a  solar  year  ? 

20.  How  many  grains  are  there  in  5  lb.  Avoirdupois  ? 

21.  A  quart  of  spirits  of  camphor  will  fill  how  many  4-ounce 
bottles  ? 

22.  What  is  the  area  of  a  triangle  whose  base  is  2  ft.  and 
whose  altitude  is  20  in.  ? 

23.  What  is  the  altitude  of  a  parallelogram  having  an  area 
of  96  sq.  in.  and  a  base  of  2  ft.  ? 

24.  A  piece  of  lumber  2'^  by  4'',  and  6  ft.  long,  contains  how 
many  board  feet  ? 

25.  a.  How  many  shingles  are  required  for  1  square  foot  of 
roof,  when  they  are  laid  6  inches  to  the  weather  ? 

h.    How  many  are  required  for  one  square  of  roofing  ? 

26.  What  is  the  cost  of  a  slate  roof  20'  x  30'  at  $10  per 
square  ? 

27.  A  grocer  sold  66|  %  of  a  hogshead  of  vinegar.  How 
many  gallons  did  he  sell  ? 

28.  33J  %  of  a  rod  is  how  many  feet  ? 

29.  What  per  cent  does  a  grocer  gain  on  celery  bought 
at  30^  a  dozen  heads,  and  sold  at  5/  a  head  ? 

30.  What  per  cent  does  a  merchant  gain  when  he  sells  two 
yards  of  cloth  for  what  three  yards  cost  ? 

31.  A  newsboy  bought  30  papers  and  sold  them  at  a  profit 
of  50%.  How  many  papers  can  he  buy  with  the  money 
received  for  the  papers  sold? 


278  GRAMMAR  SCHOOL  ARITHMETIC 

32.  How  much  commission  does  an  agent  receive  for  selling 
§1200  worth  of  goods,  when  the  rate  of  his  commission  is 
16|  %  ? 

33.  What  is  the  premium  for  insuring  a  $10,000  stock  of 
goods  for  one  fourth  of  its  value  at  2%  ? 

34.  Mr.  Wheelock's  county  tax  was  $75  when  the  county- 
tax  rate  was  5  mills  on  a  dollar.  What  was  the  assessed  valua- 
tion of  Mr.  Wheelock's  property  ? 

35.  A  tax  collector's  suretyship  bond  cost  him  $28,  at  the 
rate  of  $4  per  thousand.     What  was  the  amount  of  his  bond? 

36.  On  a  certain  day.  New  York  exchange  sold  in  Kansas 
City  at  1%  premium.  What  was  the  premium  on  a  $16,000 
draft  ? 

37.  When  exchange  on  London  is  quoted  at  4.87^,  what  is 
the  cost  of  a  draft  for  £  100  ? 

38.  When  exchange  on  Geneva  is  quoted  at  5.20,  what  is 
the  cost  in  Philadelphia  of  a  draft  on  Geneva  for  104  francs  ? 
What  is  the  face  of  a  draft  that  $100  will  buy  ? 

39.  What  is  the  face  of  a  Berlin  draft  that  can  be  bought 
for  $  240  when  exchange  is  quoted  at  96  ? 

40.  A  room  12  meters  long  is  how  many  feet  long  ?  (Think 
all  the  way  through  before  you  perform  any  operation.) 

41.  300  liters  of  oats  are  about  how  many  bushels  ? 

42.  100  liters  of  kerosene  oil  are  about  how  many  gallons  ? 

43.  About  how  many  square  meters  of  carpet  are  required  to 
cover  a  floor  2  rods  wide  and  4  rods  long  ? 

44.  What  is  the  scale  of  linear  measure  in  the  metric  system  ? 
Of  surface  measure  ?     Of  volume  measure  ? 

45.  State  your  weight  approximately,  in  kilograms. 


REVIEW   AND   PRACTICE  279 

477.     Written  , 

Solve  the  following  problems^  using  equations  wherever  they  will 
shorten  or  simplify  the  work: 

1.  Find  (a)  the  greatest  common  divisor,  and  (5)  the  least 
common  multiple  of  126,  210,  294,  and  462. 

2.  Reduce  yf^^  to  a  decimal. 

3.  Kerosene  is  80|  %  as  heavy  as  water.  If  a  gallon  of 
water  weighs  8^  lb.,  how  many  gallons  are  there  in  a  ship  load 
of  kerosene  weighing  3900  tons  ? 

4.  In  1890  there  were  166,706  miles  of  railroad  in  the  United 
States,  and  in  1900  there  were  190,082  miles.  What  was  the 
per  cent  of  increase  ? 

5.  The  copper  cent,  which  has  not  been  coined  since  1864, 
weighed  72  grains  and  was  composed  of  88  %  copper  and  12% 
nickel.  How  many  pounds.  Avoirdupois,  of  copper  were  there 
in  $100  worth  of  those  coins  ? 

6.  Find  the  number  of  gallons  of  water  that  can  be  con- 
tained in  a  rectangular  cistern  7  ft.  by  12  ft.  by  5|  ft. 

7.  On  the  29th  day  of  April,  1908,  Francis  Burns  bought 
of  Fred  J.  Peck,  9  tons  of  egg  coal  and  5  tons  of  chestnut  coal 
at  16.10  per  ton,  and  2  tons  of  pea  coal  at  $4.25  per  ton. 
Make  out  the  bill  and  receipt  it  as  the  creditor's  agent. 

8.  A  boy  spent  |  of  his  money,  earned  Qb  cents,  and  then 
had  J  of  his  original  sum.  How  much  money  had  he  at  first  ? 
(Let  X  =  the  money  he  had  at  first.) 

9.  A  man  owning  135  acres  of  land,  sold  63  A.  87  sq.  rd. 
How  much  land  had  he  left  ? 

10.  Add  40°  37'  19'^  20°  40'  30'',  and  9°  30'  45". 

11.  Divide  35°  21'  30"  by  15. 


280  GRAMMAR  SCHOOL   ARITHMETIC 

12.  How  many  cords  are  there  in  a  pile  of  4-foot  wood  7  ft. 
high  and  40  ft.  long  ? 

13.  Find  the  cost,  at  36  cents  per  square  yard,  of  plastering 
the  four  walls  and  ceiling  of  a  store  72  ft.  long,  36  ft.  wide, 
and  12  ft.  3  in.  high,  allowing  375  sq.  ft.  for  openings. 

14.  What  is  the  cost  of  carpeting  a  room  14  ft.  9  in.  long 
and  12  ft.  6  in.  wide  with  Brussels  carpet  27  in.  wide,  costing 
$1.35  a  yard,  running  the  strips  lengthwise  of  the  room  and 
making  no  allowance  for  waste  in  matching  the  pattern? 

15.  Find  the  cost  of  48  planks,  16  ft.  long,  14  in.  wide, 
and  3  in.  thick,  at  1 34  per  M. 

16.  What  is  the  altitude  of  a  triangle  whose  area  is  600 
sq.  ft.  and  whose  base  is  60  ft.  ?  (Let  x  —  the  altitude  and 
make  an  equation.) 

17.  A  building  lot  was  sold  for  f  1150,  which  was  an  advance 
of  15  (fo  on  the  cost.  If  it  had  been  sold  for  $2210,  what  would 
have  been  the  rate  per  cent  of  gain  ? 

18.  A  farm,  sold  at  a  loss  of  18%,  brought  $16,400.  How 
many  dollars  were  lost  ? 

19.  At  what  price  must  cloth  costing  $3.50  per  yard  be 
marked,  that  the  merchant  may  deduct  20  %  from  the  marked 
price  and  still  gain  20  %  ? 

20.  One  brand  of  tin  plate  is  made  by  dipping  thin  steel 
plates  into  molten  tin.  A  coating  of  tin  adheres  to  the  steel, 
making  a  sheet  of  bright  tin. 

a.  If  112  of  the  plates  weigh  98  lb.  before  being  dipped, 
and  106  lb.  after  being  dipped,  what  per  cent  of  the  tin  plate  is 
tin? 

h.    What  per  cent  of  the  tin  plate  is  steel  ? 

c.    How  many  pounds  of  tin  will  2800  tin  plates  contain  ? 


REVIEW   AND  PRACTICE 


281 


21.  The  following  is  a  record  of  receipts  and  expenses  for 
one  year  of  a  94-acre  farm  in  New  York  State,  owned  by  Mr. 
Tallcott,  and  worked  by  a  tenant  who  received  one  half  of  the 
net  income  as  his  share  : 


Receipts 

Wheat,  107  bu.,  at 

Potatoes,  598  bu.,  at 

Cabbage,  44  tons,  at 

Hay,  11  Yo  tons,  at 

Milk 

Veal 

Young  stock,  growth 

Nine  pigs 

Poultry 


80^  per  bu. 
60^  per  bu. 
$14.40  per  T. 
$11.00  per  T. 
$239.00 
$22.00 
$50.00 
^$106.00 
'  $92.00 


Expenses 

Phosphates  $47 

Seed  $23 

Miscellaneous  $94 


a.    How  much  did  the  tenant  receive  for  his  year's  work? 

h.  The  owner's  entire  investment  consisted  of  $2700  paid  for 
the  farm,  $500  for  improvements,  and  $800  for  stock.  Out  of 
his  share  of  the  profits,  he  paid  $35  taxes  and  insurance,  $68  for 
repairs,  and  $90  for  other  items.  His  net  income  was  what 
per  cent  of  his  investment  ? 

c.  The  next  year,  the  income  from  produce  (cabbages, 
wheat,  potatoes,  etc.)  diminished  $388.  The  income  from 
milk  and  live  stock  increased  $407,  and  the  expenses  increased 
$168.28.  Was  Mr.  Tallcott's  per  cent  of  net  income  increased 
or  diminished,  and  how  much  ? 

22.  How  many  steres  of  stone  are  there  in  a  stone  wall  3  m. 
long,  5  dm.  thick,  and  250  cm.  high  ? 


23. 


Write  10  dm.,  5  m.,  and  9  mm.  as  one  number. 


24.    How  many  liters  of  water  will  be  contained  in  a  vat 
which  is  3  m.  long,  25  dm.  wide,  and  200  cm.  deep? 


282  GRAMMAR  SCHOOL   ARITHMETIC 

25.  How  many  kilograms  of  water  will  a  rectangular  tin  box 
hold,  if  it  is  15  dm.  long,  25  cm.  deep,  and  1  m.  wide? 

26.  What  is  the  cost  of  goods  that  bring  1742.56,  when  sold 
at  a  gain  of  7  %  ? 

27.  A  certain  kind  of  dress  goods  shrinks  4  %  in  sponging. 
How  many  yards  should  be  purchased  for  a  suit  requiring  12 
yd.  of  sponged  cloth? 

28.  A  sloyd  class  was  composed  of  20  boys.  Each  boy  made 
a  sled  of  the  following  parts :  runners,  42  in.  long  and  4^  in. 
wide ;  three  crosspieces,  each  2J  in.  by  12  in.  ;  a  top,  12  in.  by 
28J  in.  What  was  the  cost  of  the  lumber  at  870  per  M,  none 
of  it  being  more  than  1  in.  thick,  and  estimating  that  20  %  of 
all  the  lumber  purchased  was  wasted  in  the  work? 

Hint.  —  What  per  cent  of  the  lumber  was  not  wasted? 

29.  For  what  sum  must  I  give  my  note,  without  interest,  due 
90  days  from  date,  in  order  that  it  may  yield  i 492. 50,  when 
discounted  at  6  %  on  the  day  of  date  ? 

30.  A  note  for  i600,  without  interest,  dated  July  1,  due 
90  days  from  date,  was  discounted  Aug.  30,  at  the  rate  of  7  % 
per  annum.     Find  the  proceeds. 

31.  What  .principal  will  give  $63  interest  in  2  yr.  3  mo.  at 

8%? 

32.  At  what  rate  of  interest  will  $600  amount  to  $692  in 
2  yr.  6  mo.  20  da.  ? 

33.  Find  the  interest  on  $390  for  1  yr.  6  mo.  5  da.  at  5%. 

34.  What  per  cent  of  the  list  price  is  paid  by  a  purchaser 
who  is  allowed  discounts  of  20  %  and  10  %  ? 

35.  The  premium  on  an  insurance  policy  is  $33,  and  the  rate 
To%*     What  is  the  face  of  the  policy? 


REVIEW  AND   PRACTICE  283 

36.  A  certain  village  must  raise  19017  by  taxation.  There 
are  670  men  who  pay  a  poll  tax  of  $1  each.  The  assessed 
valuation  of  the  property  of  the  village  is  1667,760. 

a.    What  must  be  the  tax  rate  ? 

h.    What  is  the  tax  on  property  assessed  at  $7500  ? 

c.  What  is  the  entire  tax  of  a  man  whose  property  is  assessed 
at  $1475  and  who  is  a  resident  of  the  village  ? 

d.  What  is  the  assessed  valuation  of  property  on  which  the 
tax  is  195.50? 

37.  In  order  to  close  out  a  stock  of  gloves  that  cost  me 
$9.60  a  dozen  pairs,  I  am  selling  them  at  $.75  a  pair.  What 
per  cent  do  I  lose  ? 

38.  A  collector  collected  a  sum  of  money,  took  out  his  com- 
mission of  3  %,  and  sent  his  principal  the  remainder,  which  was 
$3636.53.     How  much  did  he  collect  ? 

39.  Marcus  Stevens,  of  Fort  Wayne,  Ind.,  bought  of  Johnson 
&  Co.  of  Harrisburg,  Pa.,  a  bill  of  goods  amounting  to  $673. 
Johnson  &  Co.  shipped  the  goods  and  drew  on  Stevens  for  the 
amount,  at  60  days  sight,  through  the  First  National  Bank  of 
Harrisburg.  The  draft  was  accepted  by  Stevens,  discounted 
by  the  First  National  Bank  at  6%,  and  the  proceeds  credited 
to  the  account  of  Johnson  &  Co. 

a.    Write  the  draft  as  it  was  when  discounted. 

h.    What  was  the  amount  credited  to  Johnson  &  Co.? 

40.  Franklin  J.  Becker,  of  Nashville,  imported  10  cases  of 
machinery  from  Germany,  invoiced  at  7700  marks.  Find  the 
duty,  in  United  States  money,  at  45  %  ad  valorem. 

41.  a.  What  is  the  duty,  at  10^  per  gallon,  on  400  barrels 
of  rape- seed  oil,  each  barrel  containing  52  gallons  ? 

h.    If  the  oil  was  invoiced  at  69,000  francs,  what  must  be  the 


284  GRAMMAR   SCHOOL   ARITHMETIC 

cost  of  a  draft  on  Paris  sufficient  to  pay  the  bill,  the  rate  of 
exchange  being  5.17|^  ? 

STOCKS 

478.  It  often  happens  that  one  man  or  a  small  group  of  men 
desire  to  engage  in  a  business  that  requires  more  money,  or 
capital  as  it  is  called,  than  they  alone  are  able  or  willing  to 
invest  in  it.  They  obtain  more  money  by  organizing  a  stock 
company.  That  is,  they  draw  up  a  subscription  paper,  describ- 
ing the  business  in  which  they  purpose  to  engage,  the  signers  of 
which  agree,  on  certain  conditions,  to  pay  into  the  treasury  of 
the  company  the  sums  of  money  set  opposite  their  names  in  the 
subscription  paper. 

For  convenience,  the  entire  amount  to  be  raised  is  divided 
into  a  certain  number  of  parts,  called  shares,  and  each  sub- 
scriber may  subscribe  for  as  many  shares  as  he  desires. 

The  shares  of  railroad,  steamship,  telegraph,  banking,  and 
manufacturing  companies  are  usually  ilOO  each.  The  shares  of 
Western  mining  companies  are  usually  one  dollar  each.  Some- 
times shares  are  even  less  than  one  dollar. 

When  a  sufficient  number  of  shares  have  been  subscribed  for, 
the  company  is  organized,  and  receives  from  the  state  or  govern- 
ment a  charter  or  certificate  of  incorporation  empowering  it  to 
transact  business  as  an  individual.  The  shareholders  elect 
certain  ones  of  their  number,  generally  not  less  than  five,  to  be 
the  directors  of  the  company.  In  voting,  each  shareholder  has 
as  many  votes  as  the  number  of  shares  which  he  owns.  The 
directors  elect  officers  whose  duty  it  is  to  manage  the  business. 

Each  shareholder  receives  a  certificate  of  stock,  which  is  a 
document,  signed  by  officers  of  the  company,  stating  the  size  of 
each  share  and  the  number  of  shares  which  he  owns.  These 
shares  may  be  bought  and  sold  like  any  other  property. 


STOCKS 


285 


MMMMMMMMMHiMlltMMMM 


AUTHORIZED    CAPITAL 
$   23.000. 


THE  CAZENDVIA  NATIONAL  BANK 

^j^^/IS^a^w^/THE  CAZENOVIA  NATIONAL  BANK  /mMA- 


C\icJLojcr\s3 


^w. 


MiiilitsitM^MMnitittiiii 


Certificate  of  Stock. 

On  the  back  of  the  above  certificate  is  printed  the  following 
form  for  the  transfer  of  the  shares : 

For  value  received hereby  sell,  transfer,  and  assign 

to 

the  shares  of  stock  vjithin  mentioned,  and  authorize 

to  make  the  necessary  transfer  on  the  books  of  the  company. 

Witness  my  hand  and  seal  this 

day  of 19 

fc;^] 

Witnessed  by 


When  this  form  is  properly  filled  out,  the  purchaser  may 
surrender  the  certificate  to  the  company  and  receive  a  new 
one  made  out  in  his  own  name. 


286  GRAMMAR  SCHOOL  ARITHMETIC 

479.  Oral 

1.  The  certificate  on  page  285  is  for  how  many  shares  ? 

2.  Each  share  represents  how  many  dollars  of  capital 
stock  ? 

3.  What  is  the  entire  capital  of  this  bank  ? 

4.  It  is  divided  into  how  many  shares  ? 

5.  A  certain  manufacturing  company  has  a  capital  of 
f  600,000.     This  is  equal  to  how  many  shares  of  8100  each  ? 

6.  The  capital  stocjf  of  a  certain  company  is  divided  into 
2000  shares  of  $50  each.  What  is  the  entire  amount  of  its 
capital  ?  How  many  dollars  of  capital  stock  has  a  man  who 
owns  40  shares  ? 

7.  How  many  shares  of  stock  are  there  in  a  company  whose 
capital  stock  is  #200,000,  divided  into  shares  of  i25  each? 
How  many  dollars  of  this  stock  has  a  man  who  owns  50  shares  ? 

8.  What  is  the  entire  capital  stock  of  a  company  whose 
capital  is  divided  into  10,000  shares  of  $100  each  ?  How  many 
dollars  of  this  stock  has  a  man  who  owns  50  shares  ? 

9.  Name  some  stock  companies  that  transact  business  in 
your  vicinity. 

10.  If  I  own  twenty-five  50-dollar  shares  of  Pennsylvania 
R.R.  stock,  how  many  dollars  of  stock  do  I  own? 

11.  Make  a  definition  of  (a)  a  stock  company,  (6)  capital 
stock,  ((?)  a  share,  (d)  a  certificate  of  stock. 

480.  When  a  stock  company  succeeds  in  business  so  that  its 
income  is  greater  than  its  expenses,  the  profits  are  divided 
among  the  stockholders,  each  one  receiving  a  part  of  the  profits, 
according  to  the  number  of  shares  of  capital  stock  which  he 
owns. 


STOCKS  287 

In  some  companies,  if  there  are  losses  in  the  business,  they 
are  apportioned  among  the  stockholders,  each  one  contributing 
according  to  the  number  of  shares  that  he  owns. 

The  real  value  of  a  share  of  stock  begins  to  change  very  soon 
after  it  is  issued. 

If  the  business  of  the  company  is  prosperous,  so  that  there 
are  large  profits  to  be  divided  among  the  shareholders,  people 
are  anxious  to  buy  the  shares  and  are  willing  to  pay  more  for 
them  than  their  original  or  face  value.  If  the  business  is  not 
prosperous,  so  that  there  are  no  profits,  but  sometimes  losses, 
the  shareholders  are  willing  to  sell  their  shares  for  less  than 
their  original  or  face  value. 

The  abundance  or  scarcity  of  money  in  the  great  money  cen- 
ters of  the  country,  and  the  general  condition  of  business,  also 
affect  the  real  or  market  value  of  shares. 

Summary 

481.  A  stock  company  consists  of  a  number  of  persons^  organ- 
ized under  a  general  law  or  hy  special  charter.,  and  empowered  to 
transact  business  as  a  single  individual, 

482.  The  capital  stock  of  a  company  is  the  amount  named  in 
its  charter. 

The  capital  stock  nominally  represents  the  original  investment  in  the 
company,  but  is,  in  most  cases,  either  greater  or  less  than  the  present  real 
value  of  the  company's  property. 

483.  A  share  is  one  of  the  equal  parts  into  which  the  capital 
stock  of  a  company  is  divided. 

In  this  book,  a  share  will  be  considered  as  $100  of  stock  unless  otherwise 
indicated. 

484.  A  stockholder  is  a  person  who  owns  one  or  more  shares  of 
capital  stock. 


288  GRAMMAR   SCHOOL   ARITHMETIC 

485.  The  par  value  of  a  share  of  stock  is  its  original  or  face 
value  ;  the  market  value  of  a  share  of  stock  is  the  price  for  which 
the  share  will  sell  in  the  market. 

The  market  values  of  leading  stocks  fluctuate  from  day  to 
diiy,  and  are  quoted  in  the  daily  papers;  e.g.  "N.  Y.  C,  131'' 
means  that  the  stock  of  the  New  York  Central  R.R.  Co.  is 
selling  to-day  at  $131  a  share;  "Western  Union,  56"  means 
that  the  stock  of  the  Western  Union  Telegraph  Company  is 
selling  at  $56  a  share. 

486.  When  the  market  price  of  stock  is  the  same  as  the  par 
value,  the  stock  is  said  to  be  at  par ;  when  the  market  value  is 
greater  than  the  par  value,  it  is  said  to  be  above  par,  or  at  a 
premium ;  when  the  market  value  is  less  than  the  par  value,  it 
is  said  to  be  below  par  or  at  a  discount ;  e.g.  when  the  General 
Electric  Company's  stock  is  quoted  at  113,  it  is  13  %  above  par, 
or  at  a  premium  of  13%;  when  Missouri  Pacific  R.R.  stock 
is  quoted  at  47,  it  is  53%  below  par,  or  at  a  discount  of  53%. 

The  par  value  never  changes.  A  share  of  stock  that  was  originally  f  100 
is  always  $100,  though  its  market  value  may  be  more  or  less  than  $100. 
The  par  value  of  stock,  therefore,  does  not  represent  value  at  all,  but  a  cer- 
tain quantity  or  part  of  the  entire  capital  stock  of  a  company ;  just  as,  if  you 
own  100  bushels  of  wheat,  in  an  elevator  containing  100,000  bushels,  you 
own  YwuTS  P^^*  *^f  the  entire  quantity,  though  it  may  be  worth  $125,  or  only 
$60.  It  is  always  the  same  quantity  of  wheat,  whatever  may  be  its  value. 
We  should  therefore  avoid  speaking  of  a  share  as  "SlOO  ivorth  of  stock"; 
it  is  $100  o/ stock,  like  100  yards  o/ cloth,  or  100  gallons  o/oil. 

487.  Dividends  are  the  net  profits  of  a  stock  company  divided 
among  the  stockholders  according  to  the  amount  of  stock  they  own; 
assessments  are  the  losses  apportioned  among.,  and  required  to  he 
paid  hy.,  the  stockholders  according  to  the  amount  of  stock  they  own. 

Both  dividends  and  assessments  are  computed  at  a  certain  per  cent  of  the 
par  value  of  the  capital  stock ;  e.g.  if  a  company  is  capitalized  at  $  100,000, 


STOCKS  289 

and  makes  a  net  profit  of  $2000  during  one  year,  the  profit  is  2%  of 
the  par  value  of  the  stock ;  the  company  may  therefore  declare  a  dividend 
of  2  %,  and  pay  to  each  stockholder  a  dividend  of  2  %  of  the  par  value  of  his 
stock. 

488.  Stock  companies  often  issue  two  kinds  of  stock,  namely: 
Preferred  stock,  which  consists  of  a  certain  number  of  shares 

on  which  dividends  are  paid  at  a  fixed  rate,  and 

Common  stock,  which  consists  of  the  remaining  shares,  among 
which  are  apportioned  whatever  profits  there  are  remaining 
after  payment  of  the  required  dividends  on  the  preferred 
stock. 

489.  Stocks  are  generally  bought  and  sold  by  brokers,  who 
act  as  agents  for  the  owners  of  the  stock.  Brokers  receive  as 
their  compensation  a  certain  per  cent  of  thenar  value  of  the  stock 
bought  or  sold.     This  is  called  brokerage. 

The  «sual  brokerage  is  ^%  of  the  par  value  ;  e.g.  if  a  broker 
sells  10  shares  of  stock  for  me,  his  brokerage  is  ^%  of  f  1000, 
or  11.25. 

490.  Oral 

1.  How  many  dollars  of  stock  are  represented  by  fifty  f  100 
shares  ? 

2.  Explain  the  meaning  of  each  of  the  following  quotations: 
Pacific  Transportation  Co.,  57 J;  Great  Northern,  preferred, 
117|;  American  Sugar,  lOlf;  Mexican  Central,  141;  Lighting 
Co.,  188;  U.  S.  Rubber,  common,  20,  preferred,  77. 

3.  When  stock  is  quoted  at  85,  what  is  the  market  value  of 
100  shares?  What  is  the  par  value?  Is  it  at  a  premium,  or  at 
a  discount  ?     What  per  cent  ? 

4.  When  stock  is  quoted  at  1321,  what  is  the  rate  of  pre- 
mium at  which  it  sells?  What  is  the  market  value  of  two 
shares  ? 


290  GRAMMAR  SCHOOL  ARITHMETIC 

5.  When  stock  is  quoted  at  90,  what  is  the  rate  of  discount 
at  which  it  sells?  What  is  the  market  value  of  one  share? 
How  many  shares  may  be  bought  for  $  450  ?  What  will  be  the 
cost  of  1000  shares  ? 

6.  When  stock  sells  at  a  discount  of  21|9^,  what  is  the 
quotation  ? 

.  7.  What  is  the  market  value  of  one  share  of  stock  which  is 
quoted  at  120  ?  Of  8  shares  ?  How  many  shares  can  be 
bought  for  $  480  ?     For  1 1080  ?     For  $  360  ? 

8.  When  stock  is  quoted  at  75,  what  is  the  market  value  of 
one  share?  Of  4  shares?  Of  3  shares?  Of  20  shares?  How 
many  shares  can  be  bought  for  1150?  Fori  375?  For  17500? 
For  1 1500? 

9.  1 1600  will  buy  how  many  shares  of  stock  at  80  ?  At  40  ? 
At  160? 

10.  What  must  be  paid  for  100  shares  of  Rapid  Transit  R.R. 
stock  at  491? 

11.  If  I  invest  $4000  in  U.  S.  Rubber  Company's  stock  at 
20,  how  many  shares  will  I  receive? 

12.  How  many  shares  of  Union  Pacific  R.R.  stock  at  120 
can  be  purchased  by  a  woman  who  has  $3600  to  invest? 

13.  A  mining  company's  stock  is  divided  into  $1  shares. 
What  is  the  market  value  of  200  shares,  when  they  are  quoted 
at  140? 

14.  What  is  the  brokerage,  at  ^%,  on  one  share  of  the 
Columbia  Construction  Company's  stock?  If  the  stock  is 
quoted  at  105|,  what  is  the  market  value  of  one  share?  What 
will  one  share  cost  me,  including  brokerage?  If  I  buy  two 
shares,  how  much  is  my  investment? 


STOCKS  291 

15.  What  must  I  pay  for  100  shares  of  railroad  stock,  at  par, 

including  |%  brokerage? 

16.  This  morning's  paper  tells  me  that  Southern  Pacific 
R.R.  common  stock  sold  yesterday  at  18^.  If  my  broker 
sold  100  shares  of  it  for  me  at  that  figure,  and  sent  me  the  pro- 
ceeds, after  taking  out  his  brokerage  of  |^%,  how  much  per 
share  do  I  receive?  How  much  do  I  realize  from  the  sale  of 
the  100  shares? 

17.  A  broker  sold  400  shares  of  Erie  R.R.  stock  at  16.  How 
much  did  he  receive  for  it?  How  much  was  his  brokerage  at 
1%  ?  How  much  did  the  owner  of  the  stock  realize  after  pay- 
ing the  brokerage? 

18.  Mr.  Barrett  bought,  through  a  broker,  50  shares  of  Den- 
ver &  Rio  Grande  R.R.  stock  at  20|^,  paying  ^%  brokerage 
for  buying.  How  much  did  a  share  cost  him?  What  was  his 
entire  investment?  How  much  did  the  broker  receive?  How 
much  per  share  was  received  by  the  man  who  sold  the  stock, 
after  paying  his  broker?  How  much  would  he  have  received 
for  100  shares,  at  the  same  rate? 

19.  A  manufacturing  company,  having  a  capital  of  $  100,000, 
declares  a  dividend  twice  a  year.  From  Jan.  1  to  July  1  of  a 
certain  year,  its  net  profits  amounted  to  $  3000.  The  profits 
were  what  per  cent  of  the  capital  stock?  What  rate  per  cent 
of  dividends  could  the  company  declare?  What  amount  of 
dividends  did  Mr.  Scott  receive,  if  he  owned  200  shares  ?  How 
many  shares  had  a  stockholder  who  received  $  30  in  dividends  ? 
If  this  company's  net  profits  for  the  remainder  of  the  year  were 
$  5000,  what  rate  of  dividends  could  it  declare  for  that  time  ? 

20.  A  gas  company  declared  a  dividend  of  6%,  which 
amounted  in  all  to  $  60,000.     What  was  the  capital  of  the  com- 


292  GRAMMAR  SCHOOL  ARITHMETIC 

pany  ?     How  many  shares  must  a  stockholder  have  owned,  to 
receive  $  120  in  dividends  ?     How  many  dollars  of  stock  ? 

21.  If  U.  S.  Steel,  preferred,  pays  7  %  annual  dividends,  what 
are  the  dividends  on  $10,000  of  the  stock? 

22.  I  have  some  bank  stock  that  I  bought  at  200.  How 
much  did  a  share  cost  me  ?  The  stock  paid  a  10  %  dividend 
this  year.  What  did  I  receive  on  a  share  ?  What  rate  per 
cent  of  income  do  I  receive  on  my  investment  ? 

23.  A  railroad  stock  that  was  bought  at  50  pays  a  2% 
annual  dividend.  What  was  the  cost  of  10  shares  ?  What  is 
the  income  on  10  shares  ?  The  income  is  what  per  cent  of  the 
investment  ? 

24.  The  following  article  appeared  in  a  morning  paper  Jan. 

2,1908: 

"  The  Board  of  Directors  of  the  Syracuse  Rapid  Transit  Railway  Com- 
pany, at  a  meeting  held  Dec.  30,  1907,  declared  a  dividend  of  3  per  cent  on 
the  common  stock  of  the  company,  payable  Feb.  1  to  stockholders  of  record 
at  the  close  of  business  Jan.  10,  1908. 

"  The  common  stock  of  the  company  was  quoted  on  the  Syracuse  Stock 
Exchange  yesterday  at  79  bid,  and  95  asked." 

How  many  5-cent  car  fares  would  it  take  to  pay  the  dividends 
on  10  shares  of  the  Rapid  Transit  common  stock  ?  How  much 
would  10  shares  cost,  if  bought  at  the  price  bid  ?  How  much 
would  be  received  for  10  shares,  if  sold  at  the  price  asked  ? 
What  was  the  difference  between  the  asking  price  of  100  shares 
and  the  price  bid  ?  If  this  company  paid  6  %  dividends  on  its 
preferred  stock,  what  was  the  entire  income  from  10  shares  of 
preferred  and  10  shares  of  common  stock  ? 

25.  If  I  buy  stock  at  87 J,  and  after  keeping  it  for  a  time,  sell 
it  at  95|,  paying  ^%  brokerage  both  for  selling  and  buying, 
how  much  will  I  gain  on  100  shares  ? 


STOCKS  293 

26.  A  man  bought  stock  at  par  and  sold  it  six  months  later 
at  89J,  paying  ^%  brokerage  both  for  selling  and  buying. 
What  was  his  loss  on  100  shares  ? 

27.  Cut  the  stock  quotations  from  your  daily  paper,  bring 
them  to  school,  compare  them  with  quotations  of  the  same 
stocks  as  given  in  these  exercises,  and  find  the  gains  or  losses 
that  would  have  resulted  from  buying  stocks  at  the  quotations 
here  given,  and  selling  at  the  quotations  given  in  your  paper. 

28.  If  you  buy  stocks  through  a  broker,  does  the  brokerage 
add  to,  or  take  from,  the  cost  of  the  stocks  ? 

29.  If  you  sell  stocks  through  a  broker,  does  the  brokerage 
add  to,  or  take  from,  your  receipts  from  the  sale  ? 

30.  Name  all  the  things  that  are  computed  on  the  par  value. 
Can  you  think  of  anything  that  is  not  computed  on  the  par 
value  ? 

491.    Written 

The  following  quotations  are  copied  from  a  daily  paper.  Use 
them  in  solving  problems  l-lO. 

American  Cotton  Oil 
American  Woolen 
American  Sugar 
Baltimore  and  Ohio 
Brooklyn  Rapid  Transit 
Chicago,  Mil.  and  St.  Paul 
Chicago,  Northwestern 
Manhattan 

1.  Find  the  cost,  including  ^  %  brokerage,  of 

a.  150  shares  of  American  Woolen  Co. 

b.  250  shares  of  Western  tJnion  Telegraph. 

c.  300  shares  of  Manhattan  R.R. 

d.  200  shares  of  Rock  Island  R.R. 


29| 

N.  Y.  Central 

m 

l^ 

National  Lead 

39f 

lOlf 

Northern  Pacific 

116| 

81i 

People's  Gas 

80 

38| 

Rock  Island 

15| 

105| 

Southern  Pacific,  common 

73i 

137| 

Southern  Pacific,  preferred 

107i 

125 

Western  Union 

55 

294  GRAMMAR  SCHOOL  ARITHMETIC 

2.  What  will  the  seller  realize,  allowing  |  %  brokerage,  from 
the  sale  of 

a.    175  shares  of  American  Sugar  Company  ? 
h.    95  shares  of  Brookl3^n  Rapid  Transit  R.R.  ? 

c.  200  shares  of  Chicago  and  Northwestern  R.R.  ? 

d,  400  shares  of  Chicago,  Milwaukee,  and  St.  Paul  R.R.? 

3.  350  shares  of  Southern  Pacific  common  stock  are  worth 
how  much  less  than  the  same  quantity  of  Southern  Pacific  pre- 
ferred ? 

4.  How  many  shares  of  the  People's  Gas  Company  can  be 
bought  for  %  7211.25,  which  includes  \  %  brokerage  ? 

5.  A  man  realized  %  7290  from  the  sale  of  Baltimore  and 
Ohio  R.R.  stock,  paying  \^o  brokerage.  How  many  shares 
did  he  sell  ? 

6.  How  many  shares  of  New  York  Central  R.R.  stock  must 
be  sold  to  realize  $28,012.50,  brokerage  |^%  ? 

7.  My  broker  sold  for  me  90  shares  of  stock  of  the  American 
Cotton  Oil  Company,  and  bought  with  a  part  of  the  proceeds 
60  shares  of  National  Lead  stock.  He  then  sent  me  the 
remainder  of  the  money  in  the  form  of  a  New  York  draft, 
deducting  ^%  brokerage  for  selling,  \%  for  buying,  and  25^ 
exchange  for  the  draft.     What  was  the  face  of  the  draft  ? 

8.  If  I  sell  300  shares  of  Baltimore  and  Ohio  R.  R.  stock,  how 
much  must  I  put  with  the  proceeds  of  the  sale  in  order  to  buy 
an  equal  quantity  of  Northern  Pacific,  paying  \  %  brokerage 
for  each  transaction  ? 

9.  How  much  National  Lead  stock  can  be  bought  for 
13910.50,  paying  \%  brokerage? 

10.  Find  in  your  daily  paper  the  quotations  of  some  of  these 
stocks  and  compute  the  gain  or  loss  on  25  shares  bought  at  the 
rates  given  here  and  sold  at  to-day's  prices. 


STOCKS  295 

11.  What  must  be  paid  for  700  shares  of  Southern  Railway 
stock  at  13i  ? 

12.  What  must  be  paid  for  550  shares  of  Wisconsin  Central 
Railway  stock  at  15|-,  brokerage  J  %  ? 

13.  How  many  shares  of  Illinois  Central  stock  at  128|^  can 
bebought  for  $9618.75? 

14.  How  much  Railway  Steel  Spring  stock  at  26|-  can  be 
bought  for  $  18,495,  which  includes  brokerage  at  |^  %  ? 

15.  a.  What  must  be  paid,  including  brokerage  at  -1%,  for 
190  shares  of  D.  &  H.  R.R.  stock  at  150|  ? 

h.  What  does  the  seller  realize  from  the  sale  if  he  also  pays 
I  %  brokerage  ? 

16.  When  90  shares  of  stock  are  worth  $  10,125, 

a.  What  is  the  value  of  one  share  ? 

b.  What  is  the  quotation  ? 

17.  a.  How  many  dollars  of  stock  paying  4|  %  dividends 
must  I  own  in  order  to  receive  a  dividend  of  $  900  ? 

Suggestion.  —  Let  x  =  the  number  of  dollars;  then  the  statement  of 
relation  is  .04| x  =% 900. 

h.    How  many  shares  of  stock  ? 
e.    How  much  is  it  worth  at  97|  ? 

18.  I  received  in  exchange  for  an  office  building  700  shares 
of  a  bank  stock  which  was  selling  in  the  market  at  125  and 
drawing  8  %  annual  dividends. 

a.  I  received  the  equivalent  of  how  much  money  ? 

h.  How  many  dollars  of  stock  did  I  receive  ? 

c.  What  is  the  dividend  on  this  amount  of  stock  ? 

d.  That  is  what  per  cent  of  the  value  of  the  stock  ? 

Statement  of  relation  ; of  $  87,500  =  $  5600 

or,  1 87,500  a:  =  $5600. 


296  GRAMMAR   SCHOOL   ARITHMETIC 

19.  On  the  1st  of  January,  1908,  the  Faneuil  Hall  National 
Bank  paid  a  dividend  of  1|  %. 

a.  What  was  the  dividend  on  75  shares  of  the  stock  of  this 
bank  ? 

h.  How  many  shares  of  stock  are  held  by  a  stockholder  who 
receives  $  700  in  dividends  ? 

20.  At  a  certain  time  the  stock  of  the  Pennsylvania  Tele- 
phone Company  consisted  of  88,497  shares  of  $  50  each.  The 
company  paid  6  %  dividends.  What  was  the  entire  amount  of 
one  dividend? 

21.  The  Rocky  Mountain  Bell  Telephone  Company  paid 
i  142,170  in  dividends  on  $  2,369,500  of  stock. 

a.    What  was  the  rate  of  dividends  ? 

h.  If  a  stockholder  bought  his  stock  at  80,  what  is  the  rate 
of  income  on  his  investment? 

22.  The  Maryland  Coal  Company  paid  a  dividend  of  2|%, 
June  15, 1908.    What  was  the  dividend  on  1 11,000  of  the  stock  ? 

23.  At  what  price  must  stock  paying  5  %  dividend  be  bought 
that  the  buyer  may  receive  an  income  of  6  %  on  his  investment? 

24.  Stock  bought  at  120  was  sold  at  80. 
a.    What  was  lost  on  150  shares? 

h.    What  per  cent  was  lost  ? 

BONDS 

492.  A  stock  company  or  other  body  of  people^  organized  under  a 
general  law,  or  by  special  charter,  and  empowered  to  hold  property, 
and  to  act  as  an  individual,  is  a  corporation  ;  e.g.  any  stock  com- 
pany, a  city,  an  incorporated  village,  a  college,  a  church,  a 
charitable  organization  such  as  a  hospital  or  soldiers'  home. 

Corporations  and  national,  state,  county,  and  town  governments  often  find 
H  necessary  to  borrow  money  in  order  to  nieet  extraordinary  expenditures. 


BONDS  297 

For  example,  our  national  government  borrowed  vast  sums  of  money  with 
which  to  carry  on  the  Civil  War  and  the  Spanish  War,  and  later,  to  build 
the  Panama  Canal. 

States  borrow  money  with  which  to  construct  public  buildings,  highways, 
canals,  etc.  Cities,  towns,  and  counties  borrow  money  for  similar  purposes. 
Railroad  and  manufacturing  companies  borrow  money  with  which  to  extend 
their  business. 

Mention  something  for  which  your  own  city,  town,  or  village  has  borrowed 
money. 

Governments  and  corporations,  borrowing  money,  sell  their 
interest-bearing  notes  to  any  one  who  will  buy  them,  just  as  a 
man  sells  his  note  to  a  bank  when  he  borrows  money  from  the 
bank.  These  notes  are  called  bonds.  They  are  made  payable 
at  some  future  time,  usually  several  years  after  date,  the  in- 
terest to  be  paid  annually  or  semi-annually,  at  a  fixed  rate. 
Bonds,  other  than  those  of  nations,  and  of  states,  counties, 
towns,  cities,  villages,  or  other  political  divisions  of  the  country, 
are  secured  by  mortgages  on  the  property  of  the  corporations 
issuing  them. 

Bonds  are  generally  issued  in  denominations  of  $100,  $500, 
or  $1000,  just  as  paper  money  is  issued  in  denominations  of 
$1,  $5,  $20,  etc.  Occasionally  bonds  are  issued  in  denomina- 
tions smaller  than  $100,  as  was  done  with  the  Spanish  War 
bonds,  some  of  which  were  $20  bonds. 

Thus,  if  a  corporation  wishes  to  borrow  ^50,000,  it  may  issue  fifty  1000- 
dollar  bonds,  one  hundred  500-dollar  bonds,  or  five  hundred  lOO-dollar 
bonds. 

Each  bond  is  numbered  so  that  it  may  be  distinguished  from  the  other 
bonds  of  the  same  issue.  Some  bonds  are  so  drawn  that  the  owner's  name 
must  be  registered,  with  the  number  of  the  bond,  in  the  books  of  the  gov- 
ernment or  corporation  issuing  them,  so  that  the  interest  is  payable  only  to 
the  owner  or  his  order,  and  is  sent  to  him  when  due. 

Other  bonds,  like  the  one  shown  on  page  298,  have  attached  to  them  as 
many  interest  coupons  as  there  are  interest  periods.  Each  coupon  is  pay- 
able to  the  bearer,  and  bears  the  date  when  it  is  due,  so  that  the  holder  of  the 


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BONDS  299 

bond  may  collect  his  interest  when  due  by  merely  cutting  off  the  coupon 
and  presenting  it  at  the  place  named  for  payment,  or  by  depositing  it  in 
his  bank  for  collection.  The  bond  on  page  298  had  originally  thirty  interest 
coupons  attached.     The  last  three  show  in  the  form  given. 

Summary 

493.  Bonds  are  the  interest-hearing  7iotes  of  governments  and 
corporations^  given  under  seal. 

494.  Registered  bonds  are  bonds  that  are  recorded  hy  number 
in  the  name  of  the  owner^  on  the  hooks  of  the  government  or  cor- 
poration that  issued  them. 

495.  A  coupon  is  an  interest  certificate  attached  to  a  bond. 

496.  Coupon  bonds  are  bonds  to  which  interest  coupons  are 
attached. 

497.  The  face  of  a  bond  is  the  sum  mentioned  in  the  bond. 

498.  Comparisons 

1.  Shares  of  stock  represent  the  property  of  a  corporation, 
while  bonds  represent  debts  of  the  corporation ;  stockholders  are, 
therefore,  the  owners  of  the  property  of  the  corporation,  while 
bondholders  are  its  creditors. 

2.  The  income  on  shares  of  stock  is  in  the  form  of  dividends^ 
and  its  amount  fluctuates  (except  on  preferred  stock),  depending 
on  the  prosperity  of  the  corporation's  business ;  whereas  the 
income  on  bonds  is  in  the  form  of  iriterest  at  a  fixed  rate^  and 
must  be  paid,  regardless  of  the  condition  of  the  business. 

3.  The  market  value  of  bonds,  like  that  of  stocks,  fluctuates 
from  day  to  day  ;  they  may  be  at  par^  at  a  premium^  or  at  a 
discount. 

4.  Bonds  are  bought  and  sold  through  brokers  in  the  same 
manner  as  shares  of  stock,  and  at  the  same  rates  of  brokerage. 


300  GRAMMAR  SCHOOL   ARITHMETIC 

5.  The  market  values  of  bonds  are  quoted  in  the  same  way 
as  the  market  values  of  shares  of  stock ;  e.g.  ''  U.  S.  5's,  110," 
means  that  one  dollar  of  United  States  bonds  bearing  5% 
interest  is  worth  f  1.10. 

6.  The  premium,  discount,  income,  and  brokerage  on  bonds 
is  computed  on  the  par  value.  In  this  respect,  do  bonds  re- 
semble, or  differ  from  capital  stock? 

499.    Oral 

1.  What  is  the  par  yalue  of  ten  500-dollar  bonds  ? 

2.  When  selling  at  110,  what  is  their  market  value  ? 

3.  What  must  be  paid  for  five  100-dollar  bonds  when  they 
are  quoted  at  120  ? 

4.  When  bonds  are  quoted  at  80,  how  many  dollars  of  bonds 
can  be  bought  for  $400  ? 

5.  What  is  the  annual  interest  on  a  four  per  cent  500- 
dollar  bond  ?     On  a  41  %  1000-dollar  bond  ? 

6.  How  many  dollars  of  6  %  bonds  must  I  own  in  order  to 
receive  an  annual  income  of  f  1200  from  them  ?  A  semi-annual 
income  of  11200? 

7.  How  many  5  %  100-dollar  bonds  must  I  own  in  order  to 
receive  from  them  an  annual  income  of  $750?  To  receive  an 
annual  income  of  $1000?  To  receive  a  semi-annual  income 
of  $1000? 

8.  A  farmer  invested  $9000  in  railroad  bonds  at  90.  How 
many  dollars  of  bonds  did  he  buy  ?  How  many  bonds  did  he 
obtain  if  they  were  500-dollar  bonds  ? 

9.  A  speculator  invested  $1050  in  6  %  bonds  at  105.  How 
many  dollars  of  bonds  did  he  buy?  What  was  the  annual 
interest  on  them? 


BONDS  301 

10.  A  broker  bought  for  his  principal  110,000  of  railroad 
bonds  at  89|,  charging  ^%  brokerage.  What  did  the  bonds 
cost  the  principal?  What  did  the  broker  receive  for  his 
services  ? 

11.  A  broker  sold  f  10,000  of  bonds  for  his  principal  at  89J, 
charging  ^  %  brokerage.  How  much  did  the  principal  receive  ? 
How  much  did  the  broker  receive  ? 

12.  A  $500  bond  was  sold  for  -1400.  The  selling  price  was 
what  per  cent  of  the  par  value  ?  The  bond  was  sold  at  what 
per  cent  below  par?  If  this  was  the  regular  market  value, 
how  were  that  kind  of  bonds  quoted  ?  If  it  was  a  5  %  bond, 
what  was  the  annual  interest  on  the  bond  ? 

13.  A  man  invested  17800  in  bonds  at  77 1  %,  paying  |% 
brokerage.  How  many  dollars  of  bonds  did  he  buy  ?  If  they 
were  4  %  bonds,  what  was  the  annual  interest  ? 

14.  When  the  market  value  of  a  1 1000  bond  is  11030,  how 
are  the  bonds  quoted  ? 

15.  If  a  man  invests  $1200  in  7%  bonds  quoted  at  120,  how 
much  money  does  he  receive  from  them  annually? 

16.  How  many  1000-dollar  3  %  bonds  must  a  man  buy  to 
secure  an  annual  interest  of  1600?  What  will  they  cost,  if 
bought  at  90  ? 

500.    Written 

1.    a.    What  is  the  market  value  of  S40,000  of  U.  S.  2  % 
registered  bonds  due  in  1930,  when  quoted  at  104? 
b.    What  is  the  annual  interest  ? 
e.    How  many  dollars  of  these  bonds  will  f  20,800  buy  ? 

d.  What  is  the  annual  interest  on  them  ? 

e.  What  quantity  of  these  bonds  will  $35,360  buy? 
/.    What  will  be  the  yearly  interest  on  them  ? 


302  GRAMMAR  SCHOOL  ARITHMETIC 

2.  At  one  time,  U.  S.  4  %  coupon  bonds  were  quoted  at  120. 
a.    What  was  the  cost  of  $21,500  of  those  bonds? 

h.    What  interest  did  the  government  pay  annually  on  them  ? 

c.  How  many  dollars  of  bonds  could  be  bought  for  184,600? 

d.  What  interest  did  the  government  pay  annually  on  those 
bonds  ? 

3.  Milwaukee  Electric  Railway  4i  %  bonds  once  sold  at  90. 
a.    How  many  dollars  of  the  bonds  would  $81,000  buy? 

h.    What  must  be  paid  for  119,500  of  these  bonds? 

c.  What  interest  is  the  railroad  required  to  pay  annually  on 
that  amount  of  bonds? 

d.  What  amount  of  the  bonds  would  $10,800  buy? 

e.  What  interest  would  the  railroad  be  required  to  pay 
annually  on  that  amount  of  bonds  ? 

/.  A  man  invested  $63,000  in  these  bonds.  What  interest 
did  the  railroad  pay  him  annually  ? 

g.  How  much  must  be  invested  in  these  bonds  to  secure  the 
payment  of  $2700  yearly  interest  from  the  railroad  company? 

4.  A  man  bought  $198,000  of  Atchison,  Topeka,  and  Santa 
F^  R.R.  4  %  bonds  at  96|,  paying  -1  %  brokerage. 

a.    What  did  he  pay  for  the  bonds  ? 

h.  He  sold  them  at  100  J,  paying  \  %  brokerage.  How  much 
did  he  receive  for  them? 

c.  How  much  did  he  gain  by  the  speculation? 

d.  With  the  proceeds  of  the  sale,  he  bought  Allegheny  and 
Western  first  mortgage  4  %  bonds  at  98|,  paying  \  %  broker- 
age.    What  amount  of  bonds  did  he  buy? 

5.  A  man  sold  400  shares  of  stock,  yielding  2-|-%  semi- 
annual dividends,  at  102f,  and  with  the  proceeds  bought 
Toledo,  St.  Louis,  and  Western  R.R.  4%  bonds  at  T9|,  paying 
1%  brokerage  for  each  transaction.  What  amount  of  bonds 
did  he  buy  ? 


RATIO  303 

6.  On  the  18th  of  February,  1908,  the  4%  bonds  of  the 
Adams  Express  Company  were  quoted  at  88.  Make  and  solve 
four  problems  from  the  data  here  given. 

7.  Metropolitan  Street  Railway  5  %  bonds  once  sold  at  103|. 
Make  and  solve  four  problems  using  this  fact. 

RATIO 

501.  The  ratio  of  two  numbers  is  the  quotient  obtained  by  divid- 
ing one  number  by  the  other^  e.g. : 

a.  The  ratio  of  6  to  3  is  6  ^  3,  or  2. 

b.  The  ratio  of  3  to  6  is  3  -f-  6,  or  |. 

c.  The  ratio  of  11  to  7  is  11 -r-  7,  or  If 

d.  The  ratio  of  7  to  11  is  7  -^  11,  or  ^j, 

502.  Oral 

What  is  the  ratio  of 

1.  15  to  5?  7.  30  to  3?  13.  99  to  3?  19.  36  to  35? 

2.  5  to  15?  8.  3  to  30?  14.  3  to  99?  20.  35  to  36? 

3.  24  to  8?  9.  81  to  27?  15.  625  to  25?  21.  14  to  42? 

4.  8  to  24?  10.  27  to  81?  16.  25  to  625?  22.  42  to  14? 

5.  100  to  1?  11.  ltol9?  17.  7  to  17?  23.  6  to  9? 

6.  1  to  100?  12.  19tol?  18.  17  to  7?  24.  9  to  6? 

503.  Division  is  one  method  of  comparing  numbers.  By  it 
we  determine,  not  how  much  greater  one  number  is  than 
another,  but  how  many  times  as  great;  thus,  15  is  three  times  as 
great  as  5,  and  5  is  -J  as  great  as  15. 

504.  The  numbers  compared  in  determining  the  ratio  of  one 
number  to  another  are  the  terms  of  the  ratio ;  the  first  term  of  a 
ratio  is  its  antecedent ;  the  second  term  of  a  ratio  is  its  consequent ; 


304  GRAMMAR  SCHOOL  ARITHMETIC 

the  sign  (:)  of  ratio  is  the  sign  of  division  with  the  horizontal 
line  omitted;  e.g.  the  ratio  of  14  to  2  is  expressed,  14  :  2=7; 
14  is  the  antecedent,  2  is  the  consequent,  and  7  is  the  ratio. 

505.  The  antecedent  and  consequent  taken  together  are  called  a 
couplet. 

506.  The  inverse  ratio  of  two  numbers  is  the  quotient  of  the  sec- 
ond divided  hy  the  first;  e.g.  the  inverse  ratio  of  18  to  3  is 
3  -^  18  or  J.  The  quotient  of  the  first  divided  by  the  second 
is  called  the  direct  ratio. 

507.  Oral 

Name  the  antecedent  and  the  consequent  and  give  the  ratio  of 
each  of  the  following  couplets: 

1.  18:6         3.    16:64         5.    81  :  9         7.    5  :  29         9.    |  :  f 

2.  24  :  3         4.    49  :  7  6.    13  :  4         8.    3  :  ^         10.    |  :  4 

508.  Since,  in  a  direct  ratio,  the  antecedent  is  always  a 
dividend,  the  consequent  a  divisor,  and  the  ratio  a  quotient,  the 
antecedent  must  be  the  product  of  ^the  consequent  and  ratio. 
Therefore,  the  relations  of  product  and  factors  will  enable  us 
to  determine  any  one  of  these  numbers  when  the  other  two  are 
given. 

509.  Oral 

Find  the  value  of  x  in  each  of  the  following  ratios: 

1.  51  :  17  =  2:      4.    a:  :  19  =  2       7.    f  :  i  =  a;       10.  |  :  |  =  :r 

2.  35  :  a;  =  5         5.    95  :  2;  =  5       8.    a:  :  f  =  f        11.  f  :  |  =  2; 

3.  2^:4  =  3  6.    x'.U  =  l~      9.    \l:x=2     12.  ^:x  =  l\ 

13.  The  ratio  of  the  length  to  the  breadth  of  a  table  is  3.  If 
the  length  is  12  feet,  what  is  the  breadth?  Illustrate  by  a 
drawing. 


RATIO  305 

14.  The  ratio  of  the  length  to  the  breadth  of  a  city  lot  is  2. 
If  the  breadth  is  4  rods,  what  is  the  length?  Illustrate  by  a 
drawing. 

15.  The  ratio  of  the  height  of  a  boy  to  the  height  of  a  tree  is 
^.  If  the  tree  is  35  feet  high,  how  tall  is  the  boy  ?  Illustrate 
by  a  drawing. 

16.  What  is  the  ratio  of  the  length  of  a  rod  measure  to  the 
length  of  a  yard  stick  ? 

17.  What  is  the  ratio  of 

a.  One  gallon  to  one  quart? 

h.  Two  gallons  to  16  quarts? 

c.  One  bushel  to  one  pint? 

d.  Five  dollars  to  25  cents? 

e.  Eighty  cents  to  one  dollar  ? 
/.  One  gram  to  one  grain? 

g.  One  square  meter  to  one  square  decimeter? 

h.  One  cubic  inch  to  one  gallon? 

i.  ^Itoll? 

j.  One  mark  to  one  cent? 

18.  Give  two  numbers  whose  ratio  is  5.  * 

19.  Give  two  numbers  whose  ratio  is  16. 

20.  Give  two  numbers  whose  ratio  is  2J. 

21.  Give  two  numbers  whose  ratio  is  1J-. 

22.  The  ratio  of  25  miles  to  what  distance  is  12|? 

23.  The  ratio  of  what  time  to  3  months  is  4? 

24.  When  the  consequent  is  greater  than  the  antecedent, 
the  ratio  is  what  kind  of  a  number? 

25.  When  the  ratio  is  greater  than  1,  how  do  the  antecedent 
and  consequent  compare  ? 


306  GRAMMAR  SCHOOL  ARITHMETIC 

PROPORTION 

15  :  3  compares  how  with  10  ;  2? 
148  :  18  compares  how  with  12  da.  :  2  da.  ? 
f  3  :  $  21  compares  how  with  2  men  :  14  men  ? 
15  apples  :  30  apples  compares  how  with  8  lb.  :  16  lb.  ? 
The  answers  to  the  above  questions  may  be  expressed: 

15  :  3      =        10  :  2 
$48  :  $8  =12  da.  :  2  da. 
$3  :  $21=  2  men  :  14  men 
15  apples  :  30  apples  =    8  lb.  :  16  lb. 
Of  what  is  each  of  the  above  statements  composed? 

610.    An  equality/  of  ratios  is  a  proportion. 

The  first  of  the  above  proportions  is  read,  "  15  is  to  3  as  10  is  to  2."  Read 
the  others.  Let  each  pupil  in  the  class  write  three  proportions.  What 
must  be  true  of  two  ratios  that  they  may  form  a  proportion? 

511 .  Complete  the  following  proportions : 

a.  32:8  =  28:?  i.  21  ft.  :  3  ft.  =  ?  :  5^ 

^..  ^16  :  ?  =  32  :  2  j.  6^  :  60)^  =  8  lb.  :  ? 

c.  45:9  =  10:?  h  8  girls  :  16  girls  =  1 32  :  ? 

c?.  33  :  3  =  ?  :  2  Z.  3  :  ?  =  11  :  5i 

e.  42:6  =  14:?  m.  100  :  1000  =  ?  :  70 

/.  ?:3=18:9  n.  6%:  20%  =  9%:? 

g.  $12:  $6  =  6  da.  :?  da.  o.  8  mo.  :  1  year  =  $60  :  ? 

h,  12  mi.  :  24mi.  =  2hr.:  ?hr. 

512.  The  numbers  that  form  a  proportion  are  the  terms  of  the 
proportion, 

513.  The  first  and  fourth  terms  of  a  proportion  are  the  ex- 
tremes ;  the  second  and  third  terms  are  the  means  ;  e.g.  in  the 


PROPORTION  307 

proportion  49  :  7  =  350  :  50,  49  and  50  are  the  extremes  and  7 

and  350  are  the  means. 

Note. — The  sign  (::),  called  the  sign  of  proportion,  is  sometimes  used  in- 
stead of  the  sign  of  equality,  which  means  the  same. 

514.  In  any  proportion,  the  first  term  is  the  product  of  the 
second  term  and  ratio  ;  and  the  third  term  is  the  product  of 
the  fourth  term  and  ratio,  thus, 

35  :  7  =  15 :  3 
may  be  written,  7x5:7  =  3x5:3, 

and  any  proportion  may  be  written, 

2d  term  x  ratio  :  2d  term  =  4th  term  x  ratio  :  4th  term. 
Whence,  the  product  of  the  means  =  2d  term  x  4th  term  x  ratio, 
and  the  product  of  the  extremes  =  2d  term  x  ratio  x  4th  term. 
How  does  the  product  of  the  means  compare  with  the  product 
of  the  extremes  ? 

515.  Oral 

In  the  following  proportions,  verify  the  principle  established 
above,  that  the  product  of  the  means  is  equal  to  the  product  of  the 
extremes^  thus.     In  the  proportion,  15 :  5  =  12 :  4, 

The  product  of  the  means  is  5  x  12,  or  60. 

The  product  of  the  extremes  is  15  x  4,  or  60. 

1.  9:3    =6:2  3.    3:60  =  6:120  5.    7:2    =28:8 

2.  63:21  =  3:1  4.    14:28=    2:4  6.    3:9    =    9:27 

516.  Written 

1.  Complete  the  proportion,  88 :  24  =  264 :  x^  by  finding  the 
value  of  X, 


3       24 

Solution 

88a:  =  24x264.    .Why? 

=  ?^1?W^72. 

Therefore,  88  :  24  =  264  :  72.     Ans. 

IX 


308  GRAMMAR  SCHOOL   ARITHMETIC 

2.    Complete  the  proportion,  92  :  a;  =  69  :  12. 

Solution 
69a;=  92x12.     Why? 
4       4 
?l2Lll  =  iQ,     Therefore,  92:16  =  69:12.    Ans. 

Complete  the  following  proportions  : 

3.  50:2  =  12b:x  11.  $110  :  |88  =  rr:  28 

4.  4: 17  =  a;:  34  12.  10  A. :  35  A.  =$25:  a; 

5.  24::r=18:30  13.  10  yd. :  50  yd.  =  $20  :  a; 

6.  a;:10  =  21:35  14.  81 :  84  =  a;  bu. :  132  bu. 

7.  55:20=a;:28  15.  ic:5  =  f|:$3| 

8.  a;:51  =  65;39  16.  |a;:$4  =  l:| 

9.  455:  273  =  a;:  66  17.  888  f t. :  74  ft.  =  a; :  111  hr. 
10.  a;:  240  =  209:  264  18.  |:|  =  |t:a; 

PROBLEMS   SOLVED  BY  PROPORTION' 
517.    Oral 

1.  In  the  proportion,  20  :  80  =  3  :  a;,  how  does  80  compare 
with  20  ?     How  must  the  value  of  x  compare  with  3  ? 

2.  In  the  proportion,  a; :  18  =  23  :  46,  how  does  23  compare 
with  46  ?     How  does  the  value  of  x  compare  with  18  ? 

3.  If  the  proportion,  ? :  ?  =  3  :  90,  is  completed  by  supplying 
a  first  term  and  second  term,  how  must  the  second  term  com- 
pare with  the  first  term  ? 

4.  In  any  proportion,  if  the  fourth  term  is  greater  than  the 
third,  how  must  the  second  compare  with  the  first?  If  the 
fourth  term  is  less  than  the  third,  how  must  the  second  com- 
pare with  the  first  ? 


PROPORTioi*r  309 

Turn  to  §  516  and,  without  referring  to  your  answers,  tell 
whether  the  value  of  x^  in  each  proportion,  is  greater  or  less 
than  the  other  term  in  the  same  ratio. 

518.     Written 

1.  If  12  yards  of  cloth  cost  f  14,  what  will  132  yards  cost  at 
the  same  rate  ? 

Since  the  ratio  of  12  yards  to  132  yards  is  the  same  as  the  ratio  of  $14  to 
the  required  number  of  dollars,  the  numbers  in  this  problem  may  form  a 
proportion. 

Let  X  represent  the  required  number  of  dollars  and  let  it  be  the  fourth 
term,  thus, 

?   :   ?  =14yd.:a;yd. 

Then,  since  132  yards  will  cost  more  than  12  yards,  the  fourth  term  will 
be  greater  than  the  third  term  ;  therefore  the  second  term  must  be  greater 
than  the  first  term,  and  the  proportion  is 

12yd.:132yd.  =$14:$ar. 
Solving,  12  a:  =  132  x  14.     Why  ? 

,^X3ixJ4^j,4^ 

n 

Therefore,  132  yards  will  cost  1 154.     Ans. 

There  are  many  ways  of  stating  a  proportion  for  the  solution 
of  a  problem,  but  it  is  well  to  adopt  some  one  of  them,  and  use 
it  whenever  a  problem  is  to  be  solved  by  proportion. 

The  following  outline  has  been  found  helpful : 

1.  Let  the  fourth  term  he  x^  the  required  number, 

2.  Let  the  third  term  he  the  given  number  that  denotes  the  same 
kind  of  quantity  as  the  required  answer. 

3.  Determine,  by  reading  the  problem,  whether  the  answer  will 
be  greater  or  less  than  the  third  term,  and  arrange  the  other  two 
given  numbers  accordingly,  as  the  first  and  second  terms  of  the 
proportion. 

4.  Solve  the  proportion. 


310  GRAMMAR  SCHOOL   ARITHMETIC 

Solve  the  following  prohlems  hy  proportion: 

2.  At  the  rate  of  5  tons  for  131,  how  many  tons  of  coal  can 
bought  for  $217? 

3.  If  a  man  can  earn  $  217  in  43  days,  how  much  can  he 
earn  in  301  days? 

4.  Traveling  at  the  rate  of  49  miles  in  196  minutes,  in  how 
many  minutes  will  a  trolley  car  run  7  miles  ? 

5.  What  must  be  paid  for  5700  cubic  feet  of  gas  when  3800 
cubic  feet  cost  $3.61? 

6.  What   will   8   tons   of  coal  cost,  when   17|-   tons   cost 

$78.75? 

7.  How  far  will  a  train  run  in  7  hours,  at  the  rate  of  QbQ 
Km.  in  8  hours? 

8.  What  will  it  cost  to  buy  a  new  arithmetic  for  each  pupil 
in  a  class  of  19  pupils,  when  24  arithmetics  cost  $13.20? 

9.  A  messenger  boy  rode  his  bicycle  126  miles  in  7  days. 
How  far  would  he  ride  in  29  days  at  the  same  average  rate 
per  day? 

10.  Write  the  numbers  27,  18,  26,  39,  so  as  to  form  a 
proportion. 

11.  A  farmer  sowed  6  bushels  of  grain  on  4|^  acres  of 
land.  At  the  same  rate,  what  quantity  of  seed  is  required 
for  13|  acres? 

12.  If  26J  gal.  of  oil  can  be  extracted  from  |  T.  of  cotton 
seed,  how  much  oil  can  be  produced  from  375  lb.  of  seed? 

13.  Paul  earns  75^  a  day;  his  father  earns  $3.75  a  day. 
In  how  many  days  will  Paul  earn  as  much  as  his  father  earns 
in  61  days? 


PROPORTION  311 

14.  In  a  mile  foot-race,  A  gained  on  B  at  a  uniform  rate  of 
IT  ft.  in  15  sec.  If  A  finished  in  4  min.  45  sec,  he  was  how 
many  feet  ahead  of  B? 

15.  C  and  D  bought  for  §18.75  a  load  of  hay  weighing  1^ 
tons.  1200  lb.  of  the  hay  was  put  into  C's  barn  and  the 
remainder  into  D's.     How  much  should  D  pay? 

16.  If  33  bushels  of  wheat  will  make  7  barrels  of  flour,  how 
many  bushels  are  required  for  2|  barrels  at  the  same  rate? 

17.  If  I  of  a  tract  of  land  is  sold  for  13900,  what  is  |  of  the 
tract  worth  at  the  same  price  per  acre  ? 

18.  If  315  1.  of  water  fell  on  the  roof  of  my  house  during 
a  rainstorm  of  two  hours,  how  long  must  it  rain  at  the  same 
rate  in  order  that  enough  water  may  run  from  the  roof  to  fill 
a  rectangular  cistern  35  dm.  long,  3  m.  wide,  and  75  cm.  deep  ? 

19.  a.  When  exchange  on  Berlin  is  at  the  rate  of  4  marks 
for  97  cents,  what  must  be  paid  in  Baltimore  for  a  Berlin  draft 
for  3476  marks  ? 

h.  What  is  the  face  of  a  draft  that  may  be  bought  for 
$176.54? 

20.  a.  When  exchange  on  Antwerp  is  at  the  rate  of  15.525 
francs  for  |3,  what  must  be  paid  for  a  draft  for  646.75  francs? 

h.    What  is  the  face  of  a  draft  that  can  be  bought  for  f  850  ? 

21.  A  contractor  engaged  to  construct  a  sewer  two  miles 
long  for  $58,080.  How  much  has  he  earned  when  he  has 
completed  2112  feet  of  the  sewer? 

22.  If  the  interest  on  a  sum  of  money  for  one  year  is  1 360, 
what  is  the  interest  on  the  same  sum  for  15  months,  at  the 
same  rate  ? 

23.  If  1800  yield  $48  interest  in  a  certain  time,  how  large 
a  sum  will  yield  $216  in  the  same  time  at  the  same  rate? 


312  GRAMMAR  SCHOOL   ARITHMETIC 

24.  If  stock  bought  at  80  yields  6%  income  on  the  money 
invested,  what  per  cent  would  it  yield  if  bought  at  120  ? 

25.  If  a  sum  of  money  will  buy  provisions  to  last  250  sol- 
diers for  30  days,  the  same  sum  will  purchase  provisions  to 
last  75  soldiers  how  long? 

26.  How  many  yards  of  carpet  27  inches  wide  are  required 
to  cover  as  much  floor  space  as  are  covered  by  26  yards  of 
carpet  1  yard  wide  ? 

27.  If  a  train  runs  140  mi.  in  4  hr.  30  min.,  what  is  the  rate 
per  hour? 

28.  How  many  men  must  be  employed  to  accomplish  in  35 
days  what  55  men  can  accomplish  in  21  days? 

29.  Frank's  net  profit  from  a  flock  of  24  hens  for  one  year 
was  f  17.60.  How  many  hens  must  be  added  to  the  flock  in 
order  that  the  yearly  profit,  at  the  same  rate,  may  be  i44? 

PARTITIVE    PROPORTION 

519.  Separating  a  number  into  two  or  more  parts  that  have  a 
given  ratio  is  called  partitive  proportion ;  e.g.  if  the  number  bb 
is  divided  into  four  parts,  having  the  ratio  of  1,  2,  3,  and  5,  the 
parts  are  5,  10,  15,  and  25 ;  for  1  :  2  =  5  ;  10,  2  :  3  =  10  :  15, 
3  :  5  =  15  :  25. 

520.  Written 

1.    Separate  25  into  two  parts  having  the  ratio  of  2  to  3. 

Solution 
Let  2  X  represent  one  part. 
Then  3  x  will  represent  the  other  part. 

(1)  Adding,  5  a;  =  25,  the  sum  of  the  two  parts. 

(2)  Dividing  (1)  by  5,  a:  =    5  ^^ 

(3)  Multiplying  (2)  by  2,  2  .  :.  10  )  ^^^^  Take  ^  and  f  of  25. 

(4)  Multiplying  (2)  by  3,  3  x  =  15  i  ^  ^ 


PROPORTION  313 

2.    Divide  $  87  into  four  parts  having  the  ratio  of  1,  2,  5, 

and  T. 

Solution 

Let  X,  2  a:,  5  a;,  and  7  a:  represent  the  four  parts. 
(1)    Then,  adding,  15  a:  =  $  87 


Or, 

Ans.  '^^^^  ^^'  T^'  r5'  and 

xVof  ^87. 


(2)  Dividing  (1)  by  15,  a:  =  |    5| 

(3)  Multiplying  (2)  by  2,  2  a:  =  ^  llf 

(4)  Multiplying  (2)  by  5,  5  a:  =  $  29 

(5)  Multiplying  (2)  by  7,  7  a:  =  '$  40| 

3.  Divide  91  into  two  parts  that  shall  be  to  each  other  as 
3  to  4. 

4.  Divide  as  indicated : 

•  Ratio  of  Parts 

11,  13 

2,7,1 
4,  5,  1,  3 

1,  2,  3,  6 
4,  6,  7,  3 

2,  7,  1,  2,  1 
9,  8,  7,  6,  3 
1    2   fi   i  4 

-I)  ^>  O)   55  -5 

97,  83 
1,  4,  7 

Harry  earns  $  3  while 
Joe  is  earning  f  2.     How  much  per  month  does  each  earn  ? 

6.  Mr.  Olsen  and  his  two  sons  together  received  $192  on 
pay  day,  Mr.  Olsen  receiving  $4  as  often  as  each  of  his  sons 
received  $  2.     How  much  did  each  receive? 

7.  An  orchard  contained  twice  as  many  pear  trees  as  peach 
trees  and  four  times  as  many  apple  trees  as  pear  trees.  If  the 
three  kinds  of  trees  numbered  99,  how  many  were  there  of  each 
kind? 


TMB 

ER  Divided 

Number  of  Parts 

a. 

1200 

2 

h. 

3690 

3 

c. 

$923 

4 

d. 

3179 

4 

e. 

418  bu. 

4 

/. 

624  miles 

6 

^• 

12640 

5 

h. 

430 

5 

i. 

18,000 

2 

J- 

337 

3 

5. 

Joe  and  Har 

ry  earn  I  25 

a  month. 

314  GRAMMAR   SCHOOL  ARITHMETIC 

8.  A  kind  of  medicine  is  composed  of  licorice,  ipecac,  and 
muriate  of  ammonia  in  the  ratio  of  10,  3,  and  2.  In  three 
pounds  (Avoirdupois)  of  this  medicine  there  are  how  many 
grains  of  each  of  the  three  ingredients? 

PARTNERSHIP 

521 .  When  two  or  more  individuals  oivn  and  conduct  a  business 
in  common  they  are  called  partners,  and  their  association  in  busi- 
ness is  called  a  partnership. 

A  partnership  is  different  from  a  stock  company  in  that  each  partner  has 
a  voice  in  the  actual  management  of  the  business,  and  is  personally  Hable 
for  all  the  debts  of  the  firm. 

The  profits  and  losses  of  a  partnership  are  shared  by  the  part- 
ners according  to  the  amount  of  capital  that  each  has  invested 
in  the  business,  unless  by  contract  they  agree  otherwise. 

522.  Written 

1.  A,  B,  and  C  formed  a  partnership,  furnishing  f  800,  $1000, 
and  1 1200  capital,  respectively.  They  gained  1 1500.  Divide 
the  gain  among  the  partners  in  proportion  to  their  capital. 

2.  Mr.  Wilson  and  Mr.  Mead  entered  into  partnership.  Mr. 
Wilson's  capital  was  fSOOO,  and  Mr.  Mead's  12000.  They 
gained  $  1500.     What  was  each  partner's  share  of  the  gain  ? 

3.  Jones  &  Smith  were  partners  for  a  year,  with  a  capital  of 
$3000  and  $5000  respectively.  They  gained  $2000.  Find 
each  one's  share  of  the  gain. 

4.  Three  men  form  a  partnership.  A  invests  $1250,  B 
$  2000,  and  C  $  1550.  They  gain  $  1200.  What  is  each  man's 
share  of  the  gain? 

5.  Three  men  hired  a  coach  to  convey  them  to  their  homes. 
A's  home  was  20  miles  away,  B's  24  miles,  and  C's  28  miles. 
They  paid  $  24  for  the  coach.     What  ought  each  to  pay  ? 


PARTNERSHIP  315 

6.  A  cargo  of  wheat  valued  at  f  4500  was  entirely  destroyed. 
One  third  of  it  belonged  to  A,  two  fifths  to  B,  and  the  remain- 
der to  C.  What  was  each  one's  share  of  the  loss,  there  being 
an  insurance  of  $  3600  ? 

7.  A  man  fails  in  business  owing  $  15,000,  and  his  available 
means  amount  to  only  $9000.  How  much  will  two  of  his 
creditors  receive,  to  one  of  whom  he  owes  $3000  and  to  the 
other  14500? 

8.  A  and  B  gain  in  business  1 2500,  of  which  A's  share  is 
$1000  and  B's  $1500.  What  part  of  the  capital  does  each 
furnish,  and  what  is  the  investment  of  each  if  their  joint  capi- 
tal is  $  16,000  ? 

9.  A,  B,  and  C  own  $  600  worth  of  timber  land,  which  they 
divide  in  proportion  of  3,  5,  and  7.     Find  the  value  of  each  part. 

10.  A,  B,  and  C  bought  a  business  for  $  6000,  A  furnishing 
$2500  of  the  capital,  B  $1500,  and  C  the  remainder.  If  the 
value  of  the  business  increases  to  $  8000,  and  C  buys  out  A  and 
B,  how  mucli  should  he  pay  each  of  them  ? 

11.  A  man  failing  in  business  owes  $  10,800,  and  has  property 
worth  $  7200  to  be  divided  among  his  creditors  in  proportion  to 
their  claims.  How  much  will  be  received  by  a  creditor  whose 
claim  is  $180? 

12.  A  junior  partner  owns  a  -^^  interest  in  a  business,  the 
annual  net  profits  of  which  are  $90,000.  He  also  receives  a 
salary  of  $  2500  a  year. 

a.    What  is  his  annual  income  ? 

h.  A  good  concern  offers  to  buy  his  interest  in  the  business 
for  $100,000,  giving  in  payment  a  good  real  estate  mortgage 
paying  5%  interest,  and  retaining  him  in  the  business  at  a 
salary  of  $3000  a  year.  Would  his  income  be  increased  or 
diminished  by  accepting  this  offer,  and  how  much  ? 


316  GRAMMAR   SCHOOL   ARITHMETIC 

523.    When  the  capital  of  the  partners  is  not  employed  for  the 
same  time. 

Written 

1.  A  and  B  formed  a  partnership.  A  furnished  i500  for 
8  months  and  B  $600  for  10  months.  They  gained  |360. 
What  was  each  partner's  gain  ? 

Solution 

A  $500  for  8  mo.  =  |I4000  for  1  mo. 

B  $600  for  10  mo.  =    6000  for  1  mo. 

$10000 

A*s  share  =  j\  of  |360,  or  $144. 

B's  share  =  y%  of  $360,  or  $216. 

The  use  of  $500  for  8  months  is  equivalent  to  the  use  of  $4000  for  1  month; 
and  the  use  of  $600  for  10  months  is  equivalent  to  the  use  of  $6000  for 
1  month.  Consider  A's  capital  to  be  $4000  and  B's  $6000.  A's  share 
of  the  gain  =  ^ ;  B's  share  of  the  gain  =  ^q. 

2.  A  commenced  business  with  $10,000  capital.  Four 
months  later  B  put  in  $10,500.  Their  profits  at  the  end  of  a 
year  were  $5100.     What  was  each  man's  share  of  the  gain  ? 

3.  Three  persons  loaned  sums  of  money,  at  the  same  rate, 
for  which  they  received  $1596  interest.  The  first  loaned  $4000 
for  12  mo.,  the  second  $3000  for  15  mo.,  and  the  third  $5000 
for  8  mo.     How  much  interest  did  each  receive  ? 

4.  A,  C,  and  H  form  a  partnership.  A  puts  in  $8000, 
C  $5000,  H  $10,000.  A's  capital  remains  in  the  business  8 
mo.,  C's  9  mo.,  H's  12  mo.  The  net  gain  is  $6900.  Find  each 
man's  share  of  the  gain. 

5.  A  and  B  were  in  partnership  for  2  years.  A  at  first 
invested  $2000,  and  B  $2800.  At  the  end  of  9  months  A  took 
out  $700,  and  B  put  in  $500.  They  lost  in  the  two  years 
$3740.     Apportion  the  loss. 


REVIEW   AND   PRACTICE  317 

6.  A's  capital  was  in  business  6  months,  B's  7  months,  and 
C's  11  months.  A's  gain  was  $600,  B's  11400,  and  C's  1990. 
Their  joint  capital  was  17800.     What  was  each  man's  capital  ? 

7.  A  put  1600  in  trade  for  5  months,  and  B  1700  for  6 
months.     They  gained  §228.     What  was  each  man's  share  ? 

8.  April  1,  1905,  A  goes  into  business  with  a  capital  of 
$6000;  July  1,  1905,  he  takes  in  B  as  a  partner  with  a  capital 
of  $8000;  and  Oct.  1,  1906,  they  have  gained  $2900.  Find 
the  gain  of  each. 

9.  A  merchant  failed  in  business,  owing  A  $3000,  B  $1500, 
C  $2400,  and  D  $600.  His  assets  are  $5000,  and  the  expense 
of  settling  up  his  business  will  be  $500.  What  will  each  cred- 
itor receive? 

10.  A  and  B  were  in  partnership.  B  furnished  $18,000  for 
a  year,  and  his  share  of  the  gain  was  $1296.  A  invested  his 
capital  for  9  months,  and  his  share  of  the  gain  was  $1620. 
What  was  A's  capital  ? 

REVIEW  AND  PRACTICE 
524.     Oral 

1.  What  is  the  meaning  of  "  Baltimore  and  Ohio,  85f  "  ? 

2.  What  is  the  cost  of  10  shares  of  railroad  stock  at  89 J; 
brokerage  |^  %  ? 

3.  How  many  dollars  of  bonds  will  $10,500  buy,  when  they 
are  at  5  %  premium  ? 

4.  What  is  the  income  from  10  shares  of  Lighting  Company 
stock  when  it  pays  an  annual  dividend  of  4  %  ? 

5.  How  many  dollars  of  3  %  government  bonds  must  I 
own  in  order  to  receive  $30  a  year  in  interest? 

6.  What  is  the  ratio  of  480  to  48  ?     Of  48  to  480  ? 

7.  Complete  the  proportion  a; :  16  =  5  :  20. 


318  GRAMMAR   SCHOOL  ARITHMETIC 

8.  Divide  60  into  parts  having  the  ratio  of  1,  2,  and  3. 

9.  Two  boys,  A  and  B,  bought  some  oranges  for  45  cents. 
In  sharing  them,  A  took  two  oranges  as  often  as  B  took  three. 
How  much  of  the  cost  should  each  pay  ? 

10.  The  ratio  of  a  boy's  age  to  his  father's  age  is  the  ratio 
of  1  to  7.     If  the  father  is  32  years  old,  what  is  the  boy's  age  ? 

11.  What  is  the  difference  between  bonds  and  capital  stock  ? 

12.  Draw  a  horizontal  line  on  the  blackboard.  Draw  a  ver- 
tical line  cutting  off  25  %  of  the  horizontal  line.  Draw  a  line 
I  as  long  as  the  first  one. 

13.  A  grocer  sold  some  damaged  goods  for  |  of  their  cost. 
What  per  cent  did  he  lose  ? 

14.  A  farmer  sold  90  %  of  his  crop  of  potatoes  and  had  45 
bushels  left.     How  many  bushels  did  he  raise  ? 

15.  Give  the  common  fractions  equivalent  to  the  following 
per  cents:  50%,  331%,  25%,  20%,  16|  %,  66|  %,  75%, 
621%,  871%,  121%,  10%. 

16.  Frances  missed  -^-^  of  the  words  in  a  spelling  lesson. 
What  per  cent  of  them  did  she  spell  correctly  ? 

17.  On  what  base  are  profit  and  loss  computed  ? 

18.  Goods  costing  |30  were  sold  for  $40.  What  per  cent 
was  gained? 

19.  Goods  costing  $40  were  sold  for  $30.  What  per  cent 
was  lost? 

20.  I  paid  a  bill  of  1 50,  receiving  2%  discount  for  cash. 
How  much  did  I  pay  ? 

21.  I  saved  $15  by  paying  cash  for  goods,  thereby  obtaining 
a  discount  of  5  %.  What  was  the  original  amount  of  the  bill  ? 
What  was  the  net  amount  ? 


REVIEW   AND   PRACTICE  319 

22.  The  list  price  of  a  set  of  books  was  180.  The  net  price 
was  160.     What  was  the  rate  of  discount  ? 

23.  Successive  discounts  of  10  %  and  10  %  are  equivalent  to 
what  single  discount  ? 

24.  Which  of  the  following  numbers  are  composite  :  31,  49, 
51,  87,  97,  39,  51,  71  ? 

25.  What  is  the  bank  discount  on  a  note  of  $100  for  90  days 
at  6  %  ? 

26.  A  man  paid  $7.50  premium  for  insuring  his  household 
goods,  the  rate  being  75  i  per  hundred  dollars.  What  was  the 
face  of  his  policy  ? 

27.  A  merchant  had  his  stock  of  goods  insured  for  $10,000 
for  three  years,  the  rate  being  1  % .  The  agent  who  transacted 
the  business  for  the  insurance  company  received  25  %  of  the 
premium.     What  was  the  amount  of  the  agent's  commission  ? 

28.  Without  a  rule,  draw  a  line  5  decimeters  long.  Measure 
and  correct  it. 

29.  Put  your  finger  on  the  door  40  %  of  the  distance  from 
the  top  to  the  bottom. 

30.  Describe  a  board  foot. 

31.  How  many  feet  of  lumber  are  there  in  a  scantling  3''  by 
4'^  and  10  feet  long  ? 

32.  How  many  quart  cans  of  varnish  will  cover  as  much  sur- 
face as  twenty  cans  holding  a  gallon  each  ? 

33.  What  will  a  man  receive  for  a  60-day  note  for  $200, 
without  interest,  if  he  has  it  discounted  at  date,  money  being 
worth  6  %  ? 

34.  A  90-day  note,  dated  April  1,  1908,  matured  when  ? 

35.  What  is  the  meaning  of  each  of  the  following  exchange 
quotations  :  Paris  5.19  ;  Brussels  5.20  ;  Bremen  95J  ;  London 
4.868? 


320  GRAMMAR  SCHOOL   ARITHMETIC 

36.  When  exchange  on  London  is  quoted  at  4.86|,  what  is 
the  cost  of  a  bill  of  exchange  on  London  for  £  100  ? 

37.  How  may  we  tell,  without  dividing,  whether  a  number  is 
divisible  by  25  or  not  ? 

38.  How  may  we  know,  without  actual   trial,  that   24,374 
will  not  exactly  divide  2,903,076,543? 

39.  How  many  liters  are  equivalent  to  one  cubic  meter  ? 

40.  Name  some  object  that  is  as  large  as  a  liter, 

525.    Written 

1.  A  man  paid  a  certain  sum  for  a  harness,  five  times  as 
much  for  a  carriage,  two  times  as  much  for  a  horse  as  for  the 
carriage,  and  then  had  left  as  much  as  he  paid  for  the  harness. 
He  had  $340  at  first.     What  did  each  article  cost? 

2.  What  is  the  cost  of  250  shares  of  railroad  stock  at  120|, 
brokerage  |^  ? 

3.  A  man  invested  f  31,600  in  mining  stock  at  78|^,  brokerage 

a.  How  many  shares  did  he  buy  ? 

b.  What  was  his  income  when  the  stock  paid  a  dividend  of 

4.  A  man  sold  railroad  bonds  at  93|,  paying  |  %  broker- 
age. How  many  dollars  of  bonds  must  he  sell  to  realize 
$18,600? 

5.  By  proportion,  find  the  cost  of  780  barrels  of  flour,  when 
130  barrels  cost  $780. 

6.  23 J  is  the  ratio  of  42  to  what  number  ? 

7.  69  is  the  ratio  of  what  number  to  793  ? 

8.  A  man  failed  in  business  owing  $17,500.  He  had  prop- 
erty worth  $10,000,  which  was  used  in  part  payment  of   his 


REVIEW    AND   PRACTICE  321 

debts,  the  creditors  sharing  according  to  the  amounts  owing  to 
them.  How  much  did  a  creditor  receive  to  whom  the  debtor 
owed  13750? 

9.  A  man  pays  f  120  for  three  years'  insurance  on  his 
buildings,  the  policies  amounting  to  ^  of  the  value  of  the  build- 
ings, and  the  rate  being  60/  per  hundred  for  three  years. 

a.    How  much  insurance  does  he  carry  ? 

h.    What  is  the  value  of  his  buildings? 

10.  The  tax  rate  one  year  in  a  village  was  f  12J  per  $1000 
of  assessed  valuation. 

a.  What  was  the  assessed  value  of  property  which  paid  a 
tax  of  1125? 

h.  What  was  the  entire  tax  budget,  if  the  total  valuation 
was  14,000,000  ? 

11.  An  article  was  sold  for  $4.50  after  successive  discounts 
of  40  %  and  10  %  had  been  made.     Find  the  list  price. 

12.  A  merchant  can  buy  at  one  place  a  bill  of  goods  listed 
at  fl900,  receiving  successive  discounts  of  27%  and  13%. 
At  another  place  he  can  buy  the  same  goods  at  the  same  list 
price  with  a  single  discount  of  40  % .  Which  is  the  better  rate 
for  the  purchaser  and  how  much  better  ? 

13.  A  note  for  f  900  payable  at  a  bank  90  days  after  date, 
without  interest,  was  discounted  30  days  after  date,  at  the 
legal  rate. 

a.    Write  the  note,  dating  it  at  your  place  of  residence. 
h.    Compute  the  proceeds. 

14.  At  what  rate  of  interest  will  8400  earn  1 70  in 
2  yr.   6  mo.  ? 

15.  A  load  of  hay  weighing  1  T.  2  cwt.  cost  $19.80.  At  the 
same  price  per  ton,  what  was  the  cost  of  1500  lb.  of  hay  ? 
Solve  by  proportion. 


322  GRAMMAR  SCHOOL  ARITHMETIC 

16.  A,  B,  and  C  bought  a  piece  of  property  for  150,000, 
A  furnishing  112,500,  B  117,500,  and  C  the  remainder.  They 
sold  the  property  for  $69,000.  Find  each  man's  share  of 
the  gain. 

17.  A  rectangular  cellar  measures  33  ft.  by  21  ft.  and  8  ft. 
deep,  inside  measure.  The  wall  is  of  concrete  1|  ft.  thick. 
Find  the  number  of  cubic  yards  of  concrete,  allowing  4  cubic 
yards  for  openings. 

18.  Suppose  Connellsville  coal  to  be  composed  of  the  follow- 
ing substances:  carbon,  60|-%  ;  sulphur,  1%  ;  moisture,  1^%  ; 
ash,  8  %  ;  the  remainder,  volatile  combustible  matter.  In  one 
long  ton  of  such  coal  there  are : 

a.    How  many  pounds  of  carbon? 

h.    How  many  pounds  of  moisture  ? 

c.    How  many  pounds  of  volatile  combustible  matter  ? 

19.  a.  If  a  miner  receives  42  /  per  ton  for  mining  coal,  mines 
6  tons  per  day,  5  days  in  a  week,  52  weeks  in  the  year,  and 
pays  $6  per  month  for  rent  of  his  house,  how  much  per  year 
has  he  for  other  purposes  ? 

h.    What  per  cent  of  his  money  does  he  pay  for  rent  ? 

20.  Coke  is  made  from  bituminous  coal  by  heating  it  in 
ovens.  This  process  is  called  "burning"  coke.  If  three  tons 
of  coal  will  make  two  tons  of  coke,  how  much  less  will  the 
Keystone  Coal  and  Coke  Company  receive  for  50,000  tons  of 
coal  by  selling  it  at  $1.05  per  ton,  than  by  coking  it  and  selling 
the  coke  at  $2.10  per  ton  ? 

21.  A  merchant  imported  from  London  1000  sq.  yd.  of  lino- 
leum invoiced  at  Ss.  Qd.  per  square  yard. 

a.    What  was  the  cost  in  English  money  ? 
h.    What  was  the  cost  in  United  States  money,  computing 
the  exchange  value  of  £1  at  $4.1 


INVOLUTION  323 

c.  What  was  the  duty  at  20/^  per  square  yard  and  20% 
ad  valorem  ? 

d.  If  the  freight  and  other  charges  amounted   to  $28.14, 
what  was  the  total  cost  per  yard? 

e.  At  what  price   per   yard  must  the  merchant  sell  it  to 
make  a  profit  of  40  %  ? 

22.  Divide  f  17,500  among  A,  B,  C,  and  D  so  that  their 
shares  shall  be  in  the  ratio  of  4,  3,  2,  and  11. 

23.  A,  B,  and  C  purchased  an  office  building  for  $450,000. 
The  net  income  from  rents,  after  paying  all  expenses,  was 
#22,500  per  year,  in  which  each  man  shared  according  to  his 
share  of  the  investment,  B  receiving  $7500,  A  $12,500,  and 
C  the  remainder.  How  much  money  did  each  contribute 
toward  the  purchase  price  ? 

24.  How  long  must  a  sum  of  money  be  on  interest  to  gain 
$350  interest  if  it  gains  $140  in  11  months? 

25.  How  many  men  would  be  required  to  earn  in  55  days 
as  much  money  as  77  men  can  earn  in  35  days,  if  all  receive  the 
same  wages  per  day? 

INVOLUTION 

2x2  =?  3x3       =? 

2x2x2        =?  3x3x3x3=? 

2x2x2x2      =?  5x5  =? 

2x2x2x2x2   =?  5x5x5        =? 

2x2x2x2x2x2=?  5x5x5x5=? 

4  is  what  of  2  and  2  ? 

8  is  what  of  2,  2,  and  2  ? 

81  is  what  of  3,  3,  3,  and  3? 

25  is  what  of  5  and  5  ? 

2  is  what  of  4?     Of  8?     Of  16  ?     Of  32?     Of  64? 


324  GRAMMAR  SCHOOL  ARITHMETIC 

3  is  what  of  9  ?     Of  81  ? 

5  is  what  of  25  ?     Of  125  ?     Of  625  ? 

How  do  the  factors  of  4  compare  with  each  other  ?  Of  8  ? 
Of  16?  Of  32?  Of  64?  Of  9?  Of  81  ?  Of  25?  Of  125? 
Of  625  ? 

526.  The  product  of  equal  factors  is  a  power.  Which  of  the 
numbers  given  above  are  powers  ? 

527.  The  product  of  two  'equal  factors  is  a  square  ;  e.g.  4  is  the 
square  of  2 ;  9  is  the  square  of  3 ;  25  is  the  square  of  5. 

The  area  of  a  square  surface  is  the  product  of  its  length  and  breadth. 
Since  these  are  equal,  the  area  of  a  square  is  the  square  of  either  dimension. 
For  example,  the  area  of  a  square  whose  side  is  7  ft.  is  49  sq.  ft.  49  is  the 
square  of  7.  Any  number  that  is  the  product  of  two  equal  factors  is  called 
a  square  because  it  may  be  supposed  to  represent  a  square  surface  whose 
side  is  represented  by  one  of  the  two  equal  factors. 

528.  The  product  of  three  equal  factors  is  a  cube  ;  e.g.  8  is  the 
cube  of  2  ;  27  is  the  cube  of  3  ;  125  is  the  cube  of  5. 

The  contents  of  a  cubical  solid  are  equal  to  the  cube  of  one  of  its  dimen- 
sions. For  example,  125  cu.  in.  are  the  contents  of  a  cube  whose  edge  is 
5  in.  The  product  of  three  equal  factors  is  called  a  cube  because  it  may 
always  represent  the  contents  of  a  cube  whose  edge  is  one  of  the  three 
equal  factors. 

529.  The  product  of  four  equal  factors  is  called  a  fourth  power; 
the  product  of  five  equal  factors  is  called  a  fifth  power,  and  so  on ; 
e.g.  the  fourth  power  of  3  is  81,  the  fifth  power  of  2  is  32.  A 
number  is  sometimes  called  the  first  power  of  itself. 

530.  An  exponent  is  a  figure  placed  above  and  at  the  right  of  a 
number  to  show  which  power  of  the  number  is  to  be  taken  ;  e.g.  in 
the  expressions  11^  and  5^  the  2  shows  that  the  square  of  11  is 
to  be  taken,  and  the  4  shows  that  the  fourth  power  of  5  is  to  be 
taken.  112  =  121,  is  read,  The  square  of  11  is  121.  5*=  625, 
is  read,  The  fourth  power  of  5  ii  625. 


INVOLUTION  325 

531.  Finding  the  powers  of  numbers  is  involution. 

532.  Oral 

1.  Give  rapidly  the  values  of  the  following  expressions 

12;    22;    32;    42;    52;     62;     72;     82;     92;    13;     2^ ;     3^ ;    4^ ;    5^ ;     63 
73;    83;    93;    103;    122;    202;    402.    592.    992 .    9002;     2*;     2^  ;     3* 

3^;  5^;  a)^;  (f)^;  (f)^  (li)^;  (1)*;  a)M  (|)^;  my 

.32;    .53;    .23;    .2^;    .12;    .012;    (_I_)2. 

2.  What  is  the  area  of  a  square  whose  side  is  3  ft.  ? 

3.  What  is  the  area  of  a  square  whose  side  is  12  in.  ? 

4.  What  is  the  area  of  a  square  whose  side  is  5J  yd.  ? 

5.  What  are  the  contents  of  a  cube  whose  edge  is  12 
inches?     3  feet?     2  inches?     10  inches?     J  inch?     I  inch? 

6.  What  is  the  fourth  power  of  |  ? 

7.  81  is  the  square  of  what  number  ?  100  is  the  square  of 
what  number  ?  ^  is  the  square  of  what  number  ?  |  is  the 
square  of  what  number? 

8.  What  number  raised  to  the  fourth  power  equals  81  ? 

9.  What  number  raised  to  the  fifth  power  equals  32  ? 

10.  What  is  the  cube  of  4  ?    Of  1  ?    Of  0  ?     Of  i  ?     Of  ^^  ? 

11.  What  is  the  square  of  .5  ?     Of  1.2  ? 

Written 

533.  Find  the  powers  indicated : 

1.  152  7.  135  13.  (llf)2  19.  (151)2  25.  (.7|)2 

2.  332  8.  1083  14.  2.73  20.  (17 1)3  26.  (241)3 

3.  982  9.  25.32  15.  (_5_>)5  21.  (.08)3  27.  (f)^ 

4.  872  10.  4.062  16.  2.14  22.  (1.07)2  28.  (12i)3 

5.  183  11.  .8352  17.  (_2_8_)2  23.  (2.11)2  29.  (1000)3 

6.  242  12.  4.053  18.  .00352  24.  (.012)*  30.  (.33^)6 


326  GRAMMAR  SCHOOL   ARITHMETIC 

534. 


FINDING   THE    SQUARE    OF   A 

NUMBER    EXPRESSED 

BY  TWO   FIGURES 

37  = 

30  +  7     = 

t-\-u 

37  = 

30  +  7     = 

t  +  u 

259  = 

30x7+72   = 

txu  +  u^ 

111    = 

302  +  30  X  7 

t^-ht  xu 

1369  =  302+2x30  x7 +  1''^  =  t^+ 2  x  t  xu -^u^ 

From  the  above  illustration  we  may  observe 

a.  That  any  number  expressed  by  two  significant  figures 
may  be  separated  into  two  parts,  one  of  which  is  a  certain 
number  of  tens,  and  the  other  a  certain  number  of  units. 

b.  That  the  square  of  a  number  expressed  by  two  figures 
may  be  found  by  adding  the  square  of  the  tens,  twice  the  product 
of  the  tens  and  units^  and  the  square  of  the  units  ;  thus, 

43  =40  +  3 
*     .432  =  402  +  2  X  40  X  3  +  32  =  1600  +  240  +  9  =  1849 


535.    Oral 

Find  the  value  of : 

1.    212 

5.     312 

9. 

452 

13. 

252 

17. 

652 

21. 

332 

2.    222 

6.     462 

10. 

522 

14. 

342 

18. 

552 

22. 

842 

3.     412 

7.     382 

11. 

912 

15. 

732 

19. 

422 

23. 

312 

4.     442 

8.     922 

12. 

822 

16. 

612 

20. 

432 

24. 

952 

25.    What  is  the  area  of  a  square  meadow  whose  breadth  is 
62  rods  ? 

EVOLUTION 

4  =  2x2  49  =  7x7                         36  =  6x6 

9=3x3  625  =  5x5x5x5            343=7x7x7 

8  =  2x2x2  961  =  31x31                    169  =  13x13 

125  =  5x5x5  81  =  3x3x3x3  10,000  =  10  x  10  x  10  x  10 


EVOLUTION  327 

536.  Oral 

1.  2  is  what  of  4?     Of  8? 

2.  3  is  what  of  9  ?     Of  81  ? 

3.  7  is  what  of  49?     Of  343? 

4.  5  is  what  of  25?     Of  125  ?    Of  625? 

5.  How  do  the  factors  of  49  compare  ?  Of  169  ?  Of  961  ? 
Of  81?     Of  10,000?     Of  36? 

537.  One  of  the  equal  factors  that  produce  a  number  is  a  root 
of  that  number;  e.g.  2  is  a  root  of  4,  of  8,  and  of  16  ;  5  is  a  root 
of  125  and  of  625. 

538.  One  of  the  two  equal  factors  that  produce  a  number  is  the 
square  root  of  that  number  ;  e.g.  2  is  the  square  root  of  4  ;  3  is 
the  square  root  of  9  ;  5  is  the  square  root  of  25. 

539.  One  of  the  three  equal  factors  that  produce  a  number-  is 
the  cube  root  of  that  number ;  e.g.  2  is  the  cube  root  of  8;  3  is 
the  cube  root  of  27  ;  5  is  the  cube  root  of  125. 

540.  Other  roots  are  known  as  the  fourth  root,  fifth  root, 
sixth  root,  etc.,  according  to  the  number  of  equal  factors  which 
produce  the  corresponding  power  ;  e.g.  ^  is  the  fourth  root  of 
16,  the  fifth  root  of  32,  and  the  sixth  root  of  64  ;  3  is  the 
fourth  root  of  81,  the  fifth  root  of  243,  and  the  sixth  root 
of  729. 

541.  The  radical  sign  (V  )  placed  over  a  number  indicates 
that  a  root  of  the  number  is  to  be  taken. 

542.  A  small  figure  placed  within  the  radical  sign  to  indicate 
which  root  is  to  be  taken  is  called  the  radical  index.  When  the 
square  root  is  to  be  taken,  the  index  is  omitted,  the  radical 
sign   only  being   used  ;  e.g.   V625  indicates  that   the   square 


328 


GRAMMAR   SCHOOL   ARITHMETIC 


root  of  625  is  to  be  taken  ;  V256  indicates  that  the  fourth  root 
of  256  is  to  be  taken  ;  Vl728  indicates  that  the  cube  root  of 
1728  is  to  be  taken. 

543.  A  number  whose  indicated  root  can  he  exactly/  obtained  is 
a  perfect  power ;  e.g.  V9  =  3,  a/256  =  4,  ^32  =  2;  9,  256,  and 
32  are  perfect  powers. 

544.  A  number  whose  square  root  can  be  exactly  obtained  is  a 
perfect  square  ;  25,  144,  100. 

545.  A  number  whose  cube  root  can  be  exactly  obtained  is  a 
perfect  cube  ;  e.g.  8,  64,  .027,  1728. 

546.  Finding  the  roots  of  numbers  is  evolution. 

547.  Oral 

Read  the  following  expressions  and  state  the  value  of  each: 


1.   V4 

10. 

-\/1728 

19. 

V36 

28.  V8100 

37. 

V.Ol 

2.   V49 

11. 

Vl44 

20. 

^1 

29.    V1600 

38. 

V.81 

3.   -^27 

12. 
13. 

-^625 
V81 

21. 
22. 

VI 

-v/169 

30.  V4900 

39. 
40. 

v:64 

4.   ■v'125 

31.  V14400 

v:o9 

5.  -^/Te 

14. 

VIOO 

23. 

V25 

32.  V3600 

41. 

V625 

6.   ^1/81 

15. 

^343 

24. 

V196 

33.  V6400 

42. 

V.0625 

7.  V144 

16. 

V121 

25. 

V400 

34.  vOe 

43. 

V1.44 

8.   ^1000 

17. 
18. 

^64 

26. 
27. 

V900 
V2500 

35.  v.04 

36.  V.25 

44. 
45. 

V27 

9.   ^32 

■v^IOOOO 

V.027 

46.    What  two  equal  fractions  mi 
ducei?     1?     ,V?     sV?     i?     IV 

iltiplied  together  will  pro- 

47.    The  area  of  a  square  field  is  100   square   rods.     How 
long  is  it  ? 


SQUARE   ROOT  329 

48.  What  is  the  width  of  a  square  page  whose  area  is  81 
square  inches  ? 

49.  Give  the  value  of  V^  ;   V^  ;   VJf  ;   V^  ;   V^. 

50.  7  is  the  square  root  of  what  number  ?     3  ?     11  ?     12  ? 

1  ?       1  ?      _2_  ?       3  ?       _7_  ? 

2  •        8  •        11  •       ^  •        12  • 

51.  Of  what  number  is  7  one  of  the  three  equal  factors  ? 

52.  Of  what  number  is  12  one  of  the  two  equal  factors  ? 

53.  Find  one  of  the  two  equal  factors  of  121. 

54.  What  is  the  product  of  three  factors  7  ? 

55.  The  cube  root  of  64  is  how  many  times  the  square  root 
of  64? 

56.  What  is  the  number  whose  square  root  is  1  ?  2  ?  3  ? 
4?     5?     6?     7?     8?     9? 

57.  Find  the  number  whose  square  is  225. 

58.  Find  the  number  whose  square  root  is  169. 

59.  Name  all  the  integers  whose  squares  are  less  than  100. 

60.  Name  all  the  integers  whose  square  roots  are  less  than  10. 

61.  The  cube  of  4  is  the  square  of  what  number  ? 

62.  The  square  root  of  25  is  the  cube  root  of  what  number  ? 

63.  One  of  the  five  equal  factors  that  produce  a  number  is 
called  what  ? 

SQUARE  ROOT 

548.  When  a  number  is  a  perfect  square  and  contains  but 
two  or  three  figures,  its  square  root  may  be  obtained  easily  by 
inspection;  that  is,  we  may  obtain  the  square  root  mentally^ 
using  no  written  worlc.  But  to  obtain  the  square  root  of  a 
large  number,  we  generally  require  a  direct  method  that  may 
be  expressed  in  writing.  For  example,  let  it  be  required  to 
find  the  square  root  of  5329. 


330  GRAMMAR  SCHOOL  ARITHMETIC 

In  discovering  such  a  metliod  let  us  first  consider  how  a 
square  is  made  from  a  given  square  root. 

Qo'py  the  following  tahle^  filling  in  the  results  : 


12  = 

102  = 

1002= 

10002= 

22= 

202= 

2002= 

20002  = 

32  = 

302= 

3002= 

30002= 

42  = 

402= 

4002= 

40002  = 

52  = 

502= 

5002= 

50002= 

62= 

602= 

6002= 

60002= 

72  = 

702= 

7002= 

70002= 

82  = 

802= 

8002  = 

80002= 

92= 

902= 

9002  = 

90002= 

992= 

9992= 

99992= 

From  the  results  found,  we  may  generalize  as  follows : 

The  square  of  a  number  contains  twice  as  many  places,  or 
twice  as  many  less  one,  as  the  number  itself  contains,  and 

The  square  root  of  a  number  contains  as  many  places  as  the 
square  contains  periods  of  two  figures  each,  counting  from,  the  right, 
the  left-hand  period  sometimes  containing  hut  one  figure. 

V5329,  then,  contains  how  many  places? 

Let  t  represent  the  tens'  figure  and  u  the  units'  figure  of  the 
root ;  then  the  root  may  be  represented  hjt  +  u  and  its  square 

hy  ct  +  uy. 

Multiplying  as  in  section  534, 

t  +  u 
t  +  u 
t  X  u-{-u^ 
t'^  +  txu 


(t+uy  =  t^+2xtxu-hu^ 

This  may  be  illustrated  graphically  as  follows: 
Let  t  +  u  represent  the  parts  of  a  line,  thus, 


SQUARE   ROOT 


331 


t       +    u 

txu          >    u^ 

t      r  u 

Fig.  1 


Construct  a  square  on  this  line,  thus : 

This  square  contains  a  square  whose  area  is  t^, 
another  whose  area  is  u%  and  two  parts,  each  having 
an  area  equal  to  t  x  u.  The  sum  of  all  these  parts 
is  t^  +  2  X  t  X  u  +  u%  which  agrees  with  the  square 
of  i  +  M  as  found  above. 

Since  t-\-u  may  represent  any  number 
expressed  by  two  figures,  any  square  whose 
square  root  is  expressed  by  two  figures  may 

be  supposed  to  be  the  area  of  a  square 
whose  side  is  the  required  root.  This 
square  is  always  composed  of  two  oblongs 
and  two  squares,  similar  to  those  in  Fig.  1. 
20  For  example,  282=  784,  and  V784  =  28. 

Let   28,  or  20  +  8,  be   represented  by  a 
line,  ^Q         +    8    . 

Its    square,  784,  is    represented    by    the 


20 


tx  U            \     u^ 
20x8          ;     8^ 

20"            |20x8 

Fig.  2 


square.  Fig.  2. 

Whence  we  see  that  ■\/784 


V202  +  2  X  20  X  8  + 
or,  20  +  8,  or  28. 
Returning  to  the  example  with  which  we  began. 


V5329 
53-29(70+3 


V^2  -\.2  xtxu-^u^,  or  t  +  u. 
73  Ans.  12 


txu 


+  3 


70  X  2  =  140 

429 

429 -V- 140=  3 

429 

140  +  3  =  143 

000 

143  X  3  =  429 

By  trial,  we  find  that  the  greatest  number  of 
tens  whose  square  is  not  greater  than  5329  is  7 
tens,  or  70. 

Let  70  be  the  side  of  a  square.    3 1 
Its  area  is  70  x  70,  or  4900. 


2x70 


70 


+  3 


Fig.  3 


332  GRAMMAR  SCHOOL   ARITHMETIC 

We  may  call  this  4900  sq.  ft.,  sq.  in.,  or  any  other  kind  of  square  units. 

Subtracting  4900  from  5329,  we  find  that  there  are  429  square  units  remain- 
ing. If  we  make  additions  to  two  sides  of  the  square,  we  must  make  addi- 
tions whose  combined  length  is  70  x  2,  or  140  units.  If  429  square  units  are 
added,  the  width  of  the  addition  must  be  as  many  units  as  429  -r-  140,  or  3 
units,  with  a  small  remainder.  In  order  to  make  a  complete  square,  we 
must  again  add  a  small  square  3  units  long  and  wide.  The  entire  length  of 
the  three  additions  is  2  x  70  +  3,  or  143  units,  and  the  width  is  3  units,  as 
shown  in  Fig.  3.  Their  combined  area  is  143  x  3,  or  429  square  units,  the 
exact  number  necessary  to  complete  a  square  containing  5329  square  units. 
The  entire  length  of  one  side  of  this  square  is  70  +  3,  or  73  square  units. 

The  following  form  shows  the  usual  convenient  arrangement 
of  the  work  and  the  steps  required: 

53*29^73  square  root  ^'""^  *^®  greatest  square  (of  tens)  not 

-^ —    ^  greater    than   53    (hundred).      It  is  49 

^^  (hundred).     Its  square  root  is  7  (tens). 


143  429  Write  7  (tens)  in  the  root,  and  sub- 

429  tract  49  (hundred)  from  53  (hundred). 

Bring  down  29. 

Multiply  7  (tens)  by  2,  and  write  the  product,  14,  at  the  left  of  429  for  a 
trial  divisor.  (This  is  14  tens,  or  140,  but  we  omit  the  cipher  because  we 
shall  have  another  figure  to  takes  its  place.)  429  -^  140  =  3,  with  a  small 
remainder. 

Write  3  in  the  root,  annex  3  to  14,  making  143.  Multiply  143  by  3,  the 
new  figure  in  the  root,  and  write  the  product,  429,  under  429,  the  trial  divi- 
dend. If  we  subtract,  there  is  no  remainder,  which  shows  that  73  is  the  exact 
square  root  of  5329. 

If  after  the  new  root  figure  has  been  annexed  to  the  trial  divisor  and  the 
result  multiplied  by  the  new  root  figure,  a  product  is  obtained  that  is  greater 
than  the  trial  dividend,  we  must  retrace  our  work  and  take  the  next  lower 
figure  in  place  of  the  new  root  figure,  both  in  the  root  and  in  the  divisor. 

When  the  given  number  contains  only  three  figures,  we  first  find  the 
greatest  square  not  greater  than  the  left-hand  figure.     For  example, 

7-29)27_ 
4 

47  [329 
329 


SQUARE   ROOT 

How  may  we  test  the  correctness  of  our  answers  ? 

Prove  that  16  is  the  square  root  of  256. 

Prove  that  V95,481  =  309.     Prove  that  f  =  Vf 

549.     Written 

Find  the  indicated  roots  and  test  your  work : 


333 


1.  V6724 

6. 

V1521 

11. 

V1444 

16. 

V1369 

21. 

V1024 

2.  V2809 

7. 

V6561 

12. 

V7396 

17. 

V2116 

22. 

V841 

3.  V2025 

8. 

V8281 

13. 

V9025 

18. 

V576 

23. 

V324 

4.  V3844 

9. 

V3721 

14. 

V2209 

19. 

V1225 

24. 

V676 

5.  V5476 

10. 

V3249 

15. 

V6241 

20. 

V361 

25. 

V441 

26.  Find  the  side  of  a  square  whose  area  is  3969  sq.  ft. 

27.  A  square  field  contains  10  acres.  How  many  rods  long 
and  wide  is  it  ? 

28.  A  rectangular  floor  is  twice  as  long  as  it  is  wide.  Make 
a  drawing  to  represent  it.  Its  area  is  2178  square  feet.  Find 
its  dimensions. 

29.  Find  one  of  the  two  equal  factors  that  produce  7056. 

30.  A  certain  park  is  in  the  form  of  a  rectangle  12  rods  wide 
and  108  rods  long.  What  are  the  dimensions  of  a  square  field 
that  contains  the  same  number  of  acres  of  land  as  this  park  ? 

31.  A  square  park  has  an  area  of  529  sq.  rd. 
a.   What  are  the  dimensions  ? 

h.    What  are  the  dimensions  of  a  square  park  whose  area  is 
nine  times  as  great  ? 

32.  The  product  of  two  equal  factors  is  5776.  Find  the 
factors. 

33.  The  square  of  a  certain  number  is  6241.  Find  the 
number. 


334  GRAMMAR  SCHOOL   ARITHMETIC 

550.  When  a  number  whose  square  root  we  are  to  find  con- 
tains more  than  four  figures,  we  know,  by  section  548,  that  its 
square  root  contains  more  than  two  figures. 

We  may  find  the  number  of  figures  in  the  root  by  pointing  off 
the  given  square  into  periods  of  two  figures  each,  beginning  at  the 
right,  if  the  number  is  an  integer* 

We  may  find  the  left-hand  figure  of  the  root  by  taking  the  square  root 
of  the  greatest  square  not  larger  than  the  left-hand  period  in  the  square. 
This  figure  may  represent  the  number  of  hundreds,  or  thousands,  or  units 
of  any  order  above  thousands,  according  to  the  number  of  periods  in  the 
square.  Let  us  call  this  the  known  part  of  the  root,  and  the  figures  yet  to  be 
found  the  unknown  part  of  the  root.  As  we  find  the  successive  figures  of 
the  root,  the  number  of  known  figures  will  increase,  while  the  number 
of  unknown  figures  will  diminish. 

In  every  case,  we  may  represent  the  known  part  of  the  root  by  k  and  the 
unknown  part  by  u.  Thus  the  square  root  of  any  number  may  be  expressed 
by  A;  +  w  (the  known  part  plus  the  unknown  part),  and  the  number  itself 
may  be  represented  hy  k^ +  2  x  k  x  u-i-  u^.  This  is  always  true,  although 
the  known  part  of  the  root  is  always  increasing,  and  the  unknown  part  is 
always  diminishing,  as  we  obtain  the  successive  figures  of  the  root.  Like- 
wise, the  successive  remainders  may  be  represented  by  2  x  ^  x  m  +  m^  ;  and 
we  may  always  find  the  value  of  the  new  figure,  u,  approximately,  by 
dividing  the  remainder,  with  the  new  period  annexed,  by  2  x  A:  (twice  the 
part  of  the  root  already  found). 


For  example,  let  it  be  required  to  find  V40030929. 


Solution 

40-03-09-29 

16327 

square 

root. 

36 

123 

1262 


403    =2kxu  +  u^(k  =  Q0',  2k  =  120) 
369       (403-120  =  3;  120  +  3  =  123) 


3409  =2kxu+u^(k  =  6m',   2A:  =  1260) 
2524     (3409-^1260  =  2;  1260  +  2  =  1262) 


12647 


SS529  =  2k  X  u  +  u^  (k  =  6320',  2k  =  12Q40) 
88529    (88529  -^  12640  =  7  ;  12640  +  7  =  12647) 


SQUARE   ROOT 


335 


S                        kxu                         Im^' 

S                 kxu                    \  u^\ 

S           kx  u              \    u^    \        ! 
_       __!             J         ! 

1             1         Is 

1     X    1^ 

-^                *                   U  X  W  I         ! 

1                  1            1 

^                \     u     \   u\u 

6000                  j    300     !   20  !7 

Observe  that  the  values  of  k  (the  known 
part)  and  u  (the  unknown  part)  change 
each  time  a  new  figure  is  obtained. 

In  the  foregoing  solution,  all  figures  are 
omitted  until  they  are  needed.  For  ex- 
ample, the  value  of  the  first  k  is  really 
6000  (see  Fig.  4),  but  we  omit  the  last 
two  ciphers,  and  call  it  60,  which  is  all 
that  we  need  of  the  number  now. 

Summary 

To   find   the   square   root   of   an  Fig.  4 

integer  : 

1.  Point  off  the  integer  into  periods  of  two  figures  each,  begin- 
ning at  the  right. 

2.  Find  the  greatest  perfect  square  that  is  not  greater  than  the 
left-hand  period.  Subtract  it  from  the  left-hand  period  and  write 
its  square  root  at  the  right  of  the  given  integer  for  the  first  figure 
of  the  root. 

3.  Bring  down  the  next  period. 

4.  Multiply  the  part  of  the  root  already  found  (assuming  that  a 
cipher  is  annexed).,  by  2,  and  write  the  product  at  the  left  of  the 
remainder  for  a  trial  divisor. 

5.  Divide  the  remainder  (with  period  annexed}  by  the  trial 
divisor.  Write  the  quotient  in  the  root,  and  also  annex  it  to  the- 
trial  divisor,  making  the  divisor  complete. 

6.  Multiply  the  complete  divisor  by  the  new  figure  171  the  root. 
Subtract  the  product  from  the  last  remainder  (with  period  annexed} 
and  proceed  as  before  until  all  the  periods  of  the  square  have 
been  used. 

7.  When  the  remainder  (with period  annexed)  will  not  contain  the 
trial  divisor,  place  a  cipher  in  the  root,  bring  down  another  period, 
and  annex  a  cipher  to  the  trial  divisor  for  a  new  trial  divisor. 


GRAMMAR  SCHOOL  ARITHMETIC 


551.  Written 

Find  the  square 

root  : 

1.  88B6 

6. 

60,025 

11. 

235,225 

16. 

792,100  . 

2.  585,225 

7. 

41,616 

12. 

16,184,529 

17. 

30,250,000 

3.  137,641 

8. 

822,649 

13. 

5,322,249 

18. 

64,480,900 

4.  80,089 

9. 

164,836 

14. 

826,281 

19. 

43,560,000 

5.  101,761 

10. 

95,481 

15. 

788,544 

20. 

49,084,036 

THE  SQUARE  ROOT  OF   A  DECIMAL 

552.  Oral 

1.  Find  the  square  of  .2  ;  .3  ;  .8;  .9 ;  .01  ;  .05  ;  .07  ;  .12  ; 
.08;   .001;   .005;   .011;   .008. 

2.  When  we  square  a  decimal  of  one  place,  how  many  decimal 
places  do  we  obtain  in  the  square  ?  Of  two  places  ?  Of  three 
places  ?     Of  four  places  ? 

3.  The  number  of  decimal  places  in  the  square  compares  how 
with  the  number  of  decimal  places  in  its  square  root  ? 

4.  The  number  of  decimal  places  in  the  root  compares  how 
with  the  number  of  decimal  places  in  its  square  ? 

5.  Can  a  perfect  square  have  one  decimal  place  ?  Three  deci- 
mal places  ?     Seven  decimal  places  ?    Five  decimal-  places  ? 

6.  Can  any  number  be  multiplied  by  itself  so  as  to  obtain  a 
number  consisting  only  of  a  figure  in  units'  place  and  a  figure 
in  tenths'  place  ? 

553.  The  above  discussion  forms  the  basis  of  the  following 

Summary 

To  find  the  square  root  of  a  decimal : 

1.  Beginning  at  the  decimal  pointy  point  off  the  decimal^  both 
to  the  left  (in  a  mixed  decimal)  and  to  the  rights  into  periods  of 
two  figures  each. 


THE   SQUARE   ROOT   OF   A   COMMON    FRACTION        337 

2.  Find  the  square  root  as  with  integers. 

3.  .Point  off  one  decimal  place  in  the  root  for  every  two  decimal 
places  in  the  square. 

Note  1.  — If  the  given  decimal  contains  an  odd  number  of  decimal  places,  a 
cipher  must  be  annexed  to  complete  the  right-hand  period. 

Note  2.  — The  square  root  of  a  decimal  or  an  integer  that  is  not  a  perfect 
square  may  be  found  correct  to  any  desired  number  of  decimal  places  by 
annexing  decimal  periods  of  ciphers  and  continuing  the  work  of  extracting  the 
square  root. 

554.     Written 

1.  Find  the  square  root  of: 

a.    .0625  d.    .0256  g.    .00005625         /.    24.3049 

h.    .1225  e.    .007921  h.    158.76  k.    6130.89 

c.    .8836  /.    .092416  i.    29.0521  I.    .000121 

2.  Find^  correct  to  two  decimal  places.,  the  square  root  of: 

a.    .256  d.    62.5  g.    3.  j.    4.096 

h.    .5  e.   45  h.    67.3  h.    31.3 

c.    13  /.    .75  i.    172.341  I.    .016 


§Y  =  ?      3^3  x3 
77       7^7      7x7 


555.    THE  SQUARE  ROOT   OF   A   COMMON  FRACTION 

72      49* 

From  the    above   illustration,  tell  how  a   common   fraction 

may  be  squared. 

How  may  we  find  the  square  root  of  -^^  ?     Of  |f  ?     Of  \^  ? 

Of  21  ?     Of  3-8g  ? 

Summary 

To  find  the  square  root  of  a  common  fraction : 
1.    Reduce  the  given  fraction  to  lowest  terms. 


338  GRAMMAR  SCHOOL  ARITHMETIC 

2.  Extract  the  square  root  of  the  numerator  and  of  the  denomi- 
nator. 

3.  If  either  numerator  or  denominator  is  7iot  a  perfect  square^ 
change  the  common  fraction  to  a  decimal  and  find  the  square  root 
correct  to  the  required  number  of  decimal  places. 

To  find  the  square  root  of  a  mixed  number : 

1.  Change  the  mixed  number  to  an  improper  fraction, 

2.  Find  the  square  root  by  the  method  given  above. 

556.  Oral 

1.    Find  the  square  root  of :  ^% ;  f  f  ;  M ;  \U  '  U  5  7%  5  ^  5 

2  0.     18.1.    1  _9_ 

1^'  72  '  2^'  -^^le* 

557.  Written 

Find  the  square  root  of: 


1-  IM 

6. 

Iffff 

11. 

^m\ 

16. 

251 

2-  im 

7. 

e\\\%% 

12. 

H 

17. 

¥ 

3-  mi 

8. 

m 

13. 

16| 

18. 

H 

*■  IIU 

9. 

m 

14. 

f 

19. 

9iVo 

s-  tm% 

10. 

HU 

15. 

U 

20. 

If 

Perform 

the 

ope', 

rations  indicated : 

3                                25. 

\/729 

21.    V3.26 

X  . 

,006; 

V35721-^- 

22.    Vf+f  26.    V3.532-f-6.28 

23  3      ^    rmS6  27.    V625  +  1296 

V5184     ^129600  28.    V625-f-Vl296 


24.    V4489  X  V961  29.    V25  x  16  x  81 


EVOLUTION  BY  FACTORING  339 


30.    V961  -  529  ,^     33  x  VJT 

34. 


31.  V25xVr6xV81  V41xl65 

32.  V324xV441  7xVr764 

35. —• 

33.  V961-V529  V169X7 

EVOLUTION  BY  FACTORING 

558.  The  square  root  of  a  perfect  square,  the  cube  root  of  a 
perfect  cube,  or  any  root  of  the  corresponding  perfect  power 
may  be  found  by  factoring. 

To  determine  the  method  of  evolution  by  factoring,  and  the 
reason  for  it,  let  us  study  the  relation  between  the  factors  of 
a  number  and  the  factors  of  the  square  of  that  number. 

42  =  2  x  3  X  7;  therefore  422  =  (2  x  3  x  7)2 ^ 

2x3x7x2x3x7,  or  1764. 

We  observe  that  every  factor  of  42  occurs  twice  in  the  square  of  42. 
Likewise,  every  factor  of  any  number  occurs  twice  in  the  square  of  that 
number,  three  times  in  its  cube,  four  times  in  its  fourth  power,  and  so  on. 

Conversely,  VI764  =  V2x2x3x3x7x7  =  2  x  3  x  7,  or 
42.  

Likewise  V225  =  V3  x  3  x  5  x  5  =  3  x  5,  or  15. 


V216  =V2x2x2x3x3x3  =  2x3,  or6. 

Summary 

1.  The  square  root  of  a  perfect  square  may  he  found  hy  factoring 
the  square  and  multiplying  together  one  out  of  every  pair  of  equal 
prime  factors  fo^md  in  it. 

2.  The  cube  root  of  a  perfect  cube  may  be  found  hy  factoring 
the  cube  and  multiplying  together  one  of  every  three  equal  prime 
factors  found  in  it. 


340 


GRAMMAR   SCHOOL   ARITHMETIC 


How  may  the  fourth  root  of  a  perfect  fourth  power  be  found? 
How  may  the  fifth  root  of  a  perfect  fifth  power  be  found? 

559.    Written 

Find^  hy  factoring^  the  values  of  the  following  : 


1. 

V3600 

2. 

VIOO 

3. 

V441 

6.  V1089 

7.  VT84 

8.  V7i296 


11.  Vl26xl4      16.  V2x75x6 


12.  V98  X  8 


17.  Vl8x  45x10 


13.  V32xl8        18.   ^^M 


56  2  5 


4.  Vi225       9.   V20.25       14.   V40xl0         19.   V48400 

5.  V484      10.  V72401      15.  V45xl25       20.  VI1025 


APPLICATIONS  OF  SQUARE  ROOT 

560.  A  triangle  that  contains  a  right  angle  is  a  right  triangle. 

561.  The  side  opposite  the  right  angle  in  a  right  triangle  is  the 
hypotenuse  of  the  right  triangle. 

562.  The  two  sides  that  form  the  right  angle  of  a  right  triangle 

are  the  legs  of  the  right  triangle. 


563.    When  a  right  triangle  rests  upon  one  of 
?,  the  leg  upon  which  it  rests  is  called  the 
the  other  leg  is   called  E 

^    the  perpendicular  of  the  right  tri- 
^^    £^e    Cf     angle. 

Eight  Tkianglb 

In  triangle   ABC,  which   lines  are    j{, 
the  legs  ?     In  triangle  DEF"^     In  triangle  KLM? 

In   triangle  DEF,  which  line  is  the  hypotenuse? 
In  triangle  KLM'i  Li 


APPLICATIONS   OF   SQUARE   ROOT 


341 


564.    By  geometry  it  is  proved  that 

The  square  of  the  hypotenuse  of  a  right  triangle  is  equal  to 
the  sum  of  the  squares  of  the  two  legs. 

The  truth  of  this  proposition  may  be  shown  in  many  ways,  one  of  which 
is  the  following : 


a2 

a 

6       X. 

L 

h' 

y      4 


3 


Fig.  1  Fig,  2 

Let  KLM\iQ  a  right  triangle  of  any  shape  and  W-  and  a^  of  Fig.  2  equal 
respectively  to  IP"  and  a^  of  Fig.  1.  Take  the  point  0,  in  Fig.  2,  so  that  the 
line  NO  will  be  equal  to  the  line  KL^  in  Fig.  1,  and  draw  OF  and  OR. 

In  every  case  the  triangles  1  and  2  may  be  placed  in  the  position  of  3 
and  4,  making  a  square  equal  to  x^  of  Fig.  1.  Verify  this  for  yourself  by 
cutting  the  figures  from  paper,  using  various  lengths  for  a  and  6. 

565.    From  the  foregoing  proposition  it  follows  that 

When  the  legs  of  a  right  triangle  are  hnown^  the  hypotenuse 

may  he  found  hy  addiyig  the  squares  of  the  two  legs  and  extracting 

the  square  root  of  the  sum  ;  and  that 

When  either  leg  and  the  hypotenuse  are  Tcnown^  the  other  leg 

may  he  found  hy  suhtracting  the  square  of  the  known  leg  from 

the  square  of  the  hypotenuse  and  extracting  the  square  root  of  the 

difference. 


342 


GRAMMAR  SCHOOL   ARITHMETIC 


566.    Written 

Note.  — Approximate  roots  should  be  carried  to  two  decimal  places. 

1.    Find  the  value  of  x  in  figures  A,  B,  C,  i>,  U,  F,  Cr,  H^ 
I,  and  e7. 


2.  A  rectangular  park  is  32  rods  by  24  rods.  A  walk  ex- 
tends diagonally  across  the  park,  connecting  opposite  corners. 
How  long  is  the  walk?     (Make  a  drawing.) 

3.  One  side  of  a  rectangular  field  is  68  rods.  The  diagonal 
distance  between  opposite  corners  is  85  rods.  Find  the  other 
three  sides. 

4.  One  side  of  a  rectangle  is  69  feet.  The  diagonal  of  the 
rectangle  is  115  feet.     Find  the  perimeter  of  the  rectangle. 

5.  The  area  of  a  square  is  169  square  inches,  a.  What  is 
the  length  of  one  side  ?  h.  What  is  the  length  of  its  diagonal  ? 
e.  Draw  the  square,  exact  size,  on  the  blackboard,  and  verify 
your  work  by  measuring  the  diagonal. 


APPLICATIONS  OF   SQUARE  ROOT 


343 


6.  Find  the  perimeter  of  a  square  whose  area  is  4489  sq.  ft. 

7.  Find  the  diagonal  of  a  square  whose  area  is  324  square 
inches.  Verify  your  work  by  drawing  the  square,  exact  size, 
and  measuring  the  diagonal. 

8.  a.  What  is  the  area  of  a  square  whose  perimeter  is  228 
centimeters  ?     5.    Find  its  diagonal,  correct  to  millimeters. 

9.  Draw  a  rectangle  whose  length  is  twice  its  width.  Sup- 
pose that  its  area  is  450  square  inches,  a.  What  is  its  width  ? 
h.    What  is  its  length  ?     c.    What  is  its  diagonal  ? 

10.  Three  city  streets  intersect  in  such  a  way  as  to  inclose 
a  right  triangle,  ABQ.  The  right  angle  is  at  B,  The  side 
AB  is  8.4  meters  and  the  side  BC  i^  11.2  meters.  If  two 
boys  start  at  B  and  walk  around  the  triangle  in  opposite  direc- 
tions at  the  same  speed,  on  which  side  will  they  meet,  and  how 
far  from  A  and  from  Q  2 

11.  This  cut  represents  the  gable  end 
of  a  barn.  The  ridge  of  the  roof  is  11  ft. 
3  in.  higher  than  the  plates  on  which  the 
rafters  rest.  The  rafters  extend  18  in. 
beyond  the  plates.  How  long  must  the 
rafters  be  made  ? 

12.  Rafters  that  extend  14  in.  over  the  plates  are  21  ft.  2  in. 
long,  and  the  ridge  is  12  ft.  above  the  level  of  the  plates.  How 
wide  is  the  building  ? 

13.  How  long  a  ladder  is  needed  to  reach  a  window  24  feet 
from  the  ground,  when  the  foot  of  the  ladder  is  10  feet  from 

the  side  of  the  building  ? 

14.  This  cut  represents  the  end  of  Fred's 
chicken  house.  The  roof  extends  6  inches  over 
each  side.  Find  the  slant  height  of  the  roof,  cor- 
rect to  the  nearest  hundredth  of  a  foot. 


111495,. 


t<> 


344  GRAMMAR   SCHOOL   ARITHMETIC 

15.  a.  Measure  the  length  and  breadth  of  your  schoolroom. 
Compute  the  diagonal  of  the  floor  ;  verify  by  measurement. 

h.  Beginning  at  one  end  of  this  diagonal,  measure  the 
height  of  the  room.  What  kind  of  an  angle  is  formed  by 
the  diagonal  and  the  line  last  measured  ?  Compute  the  dis- 
tance from  the  top  of  that  line  to  the  farther  end  of  the 
diagonal. 

16.  If  a  chalk  box  is  6  in.  long,  4  in.  wide,  and  4  in.  high, 
what  is  the  distance  from  an  upper  corner  through  the  center  of 
the  box  to  the  opposite  lower  corner  ? 

17.  Find  the  perimeter  of  a  right  triangle  whose  legs  are 
7  ft.  and  5  ft. 

18.  What  is  the  side  of  a  square  field  containing  10  acres  ? 
Hint.  — Reduce  10  A.  to  square  rods.     Why  ? 

19.  A  baseball  diamond  was  90  ft.  square.  The  ball  was 
batted  directly  over  second  base  and  caught  by  a  fielder  who 
stood  90  ft.  from  second  base.  How  far  from  the  home  plate 
did  he  stand  ? 

20.  What  is  the  side  of  a  square  field  containing  2|-  acres  ? 

21.  What  is  the  diameter  of  the  largest  wheel  that  will  go 
through  a  rectangular  window  42  inches  by  31|  inches  ? 

22.  What  is  the  length  of  the  longest  straight  stick  that  can 
be  inclosed  in  a  box  4  in.  by  3  in.  by  7  in.  ? 

23.  A  30-acre  rectangular  field,  three  times  as  long  as  it  is 
wide,  is  bounded  on  one  side  and  one  end  by  the  highway. 
How  much  distance  will  a  traveler  save  by  going  in  a  direct 
line  diagonally  across  this  field,  from  corner  to  corner,  instead 
of  following  the  highway? 

MENSURATION 

Review  measurement  of  surfaces  and  rectangular  solids,  pages  96-100. 


PLANE  FIGURES 


345 


PLANE    FIGURES 

567.  A  plane  surface  is  a  surface  such  that  if  any  two  points 
in  it  are  connected  hy  a  straight  line,  the  straight  line  will  lie  wholly 
in  the  surface  ;  e.g.  a  table  top,  the  surface  of  a  window  pane. 
Test  these  and  other  surfaces  by  a  thread  held  taut. 

568.  A  portion  of  a  plane  surface  hounded  hy  lines  is  a  plane 
figure  ;  e.g.  'a.  square,  a  triangle,  a  circle. 

569.  A  plane  figure  hounded  hy  straight  lines  is  a  polygon. 

A  polygon  of  three  sides  is  called  what  ?     A  polygon  of  four  sides  ? 


570.  A  polygon  of  five  sides  is  a  pentagon  ;  of  six  sides  a  hexa- 
gon; of  seven  sides,  a  heptagon  ;  of  eight  sides,  an  octagon. 

AREAS  OF  REGULAR  POLYGONS 

571.  A  polygon  whose  sides  are  equal  and  whose  angles  are 
equal  is  a  regular  polygon  ;  e.g. 


572.    The  area  of  any  regular  polygon  may  he  found  hy  dividing 
the  polygon  into   as  many   equal  triangles  as 
the  polygon   has    sides,    and    multiplying   the 
area    of    one     triangle     hy     the    numher    of 
triangles;    e.g. 

The  area  of  this  regular  hexagon  is  six  times  the 
area  of  one  of  the  triangles,  or  six  times  one  half  of 
the  product  of  a  and  h. 


346 


GRAMMAR   SCHOOL   ARITHMETIC 


AREAS   OF   TRAPEZOIDS 

573.    A  quadrilateral  having  two  and  only  two  sides  parallel  is  a 
trapezoid. 


1/ 

/ 

^\' i  / 

/  % 

yC 

A    / 

/  ^^\ 

Tea 

'EZOIDS 

Alt. 


In  each  of  the  above  figures,  how  does  the  part  A  compare 
with  the  part  B  ? 

How  does  the  area  of  the  trapezoid  compare  with  that  of  the 
parallelogram  which  is  made  from  the  trapezoid  ?  How  is  the 
area  of  the  parallelogram  found  ? 

Observe  that  in  each  figure  the  base  of  the  parallelogram 
is  equal  to  one  half  of  the  sum  of  the  parallel  sides  of  the 
trapezoid. 

Summary 

7^e  area  of  a  trapezoid  is  equal  to  one  half  of  the  sum  of  the 
parallel  sides  multiplied  hy  the  altitude. 

574.    Written 

1.  Draw  a  trapezoid  whose  altitude  is  13  inches  and  whose 
parallel  sides  are  17  inches  and  19  inches.     Find  its  area. 

2.  Find  the  area  of  a  trapezoid  whose  parallel  sides  are  20 
feet  and  25  feet,  and  whose  altitude  is  15  feet. 

3.  A  field  in  the  form  of  a  trapezoid  has  two  parallel  sides 
of  30  rods  and  35  rods  ;  the  distance  between  them  is  20  rods. 
How  many  acres  of  land  does  the  field  contain  ? 

4.  A  board  is  1  inch  thick,  12  feet  long,  11  inches  wide  at 
one  end  and  a  foot  wide  at  the  other  end.  How  many  board 
feet  does  it  contain  ? 


STUDY  OF   THE  CIRCLE  347 

5.  A  vineyard  in  France  is  in  the  form  of  a  trapezoid,  of 
which  the  two  parallel  sides  are  185  meters  and  155  meters, 
and  the  altitude  is  130  meters.  b         42'      c 

a.    It  has  an  area  of  how  many  ares? 

h.    How  many  hectares  ? 


6.  Find  the  area  of  trapezoid  ABOD.  a  b 

7.  The  parallel  sides  of  a  trapezoid  are  41  cm.  and  bb  cm. 
Its  area  is  1296  sq.  cm.     What  is  its  altitude  ? 

Let  X  =  the  altitude. 

8.  The  area  of  a  trapezoid  is  560.5  sq.  ft.     The  altitude  is 
19  ft.     The  difference  of  the  parallel  sides  is  5  ft. 

a.    Find  the  sum  of  the  parallel  sides. 

h.    Find  the  length  of  each  of  the  parallel  sides. 

STUDY  OF   THE  CIRCLE 
575.    A  plane  figure  hounded  hy  a  curved  line,  all  points  of 
which  are  equally  distant  from  a  point  within,  called  the  center,  is 
,;^T.^«my£rej,^^  a  circle. 

576.  The  boundary  line  of  a  circle  is 
the  circumference. 

577.  A  straight  line  passing  through 
the  center  of  a  circle  and  terminating 
in  the  circumference  is  the  diameter. 

578.  A  straight  line  drawn  from  the  cen- 
ter to  the  circumference  of  a  circle  is  its  radius. 

579.  It  is  proved,  by  geometry,  that  the  circumference  of  every 
circle  is  3.1416  times  its  diameter. 

580.  Oral 

1.  The  radius  of  a  circle  is  what  part  of  its  diameter  ? 

2.  What  is  the  radius  of  a  circle  whose  diameter  is  80  cm.  ? 


348  GRAMMAR  SCHOOL   ARITHMETIC 

3.  What  is  the  diameter  of  a  circle  whose  radius  is  35  cm.  ? 

4.  What  is  the  circumference  of  a  circle  whose  diameter  is 
1  foot  ? 

5.  What  is  the  circumference  of  a  circle  whose  diameter  is 
100  inches  ? 

6.  What  is  the  circumference  of  a  circle  whose  radius  is 
5  inches  ? 

7.  What  is  the  diameter  of  a  circle  whose  circumference  is 
31.416  inches  ? 

8.  What  is  the  radius  of  a  circle  whose  circumference  is 
3.1416  meters '-^ 

Written 

1.  What  is  the  circumference  of  a  circle  whose  diameter  is 
50  inches  ? 

2.  What  is  the  radius  of  a  circle  whose  circumference  is 
182. 2128  feet? 

3.  What  is  the  diameter  of  a  circle  whose  circumference  is 
7854  miles? 

4.  The  radius  of  the  earth  is  approximately  4000  miles. 
What  is  its  approximate  circumference  ? 

5.  The  diameter  of  my  bicycle  wheels  is  28  inches. 

a.  How  many  feet  will  I  travel  during  700  rotations  of 
a  wheel  ? 

h.    How  many  meters  will  I  travel  ? 

c.  How  many  rotations  will  a  wheel  make  in  traveling 
1  mile? 

6.  A  horse  is  tethered  to  a  stake  by  a  rope  50  ft.  long.  What 
is  the  circumference  of  the  circle  over  which  he  can  graze  ? 


STUDY   OF   THE   CIRCLE 


349 


7.  The  wire  cable  of  a  hoisting 
apparatus  winds  upon  a  cylindrical 
steel  drum  20  inches  in  diameter  and 
3  feet  long.  How  many  feet  of  cable 
will  the  drum  hold,  when  wound  full, 
if  the  cable  is  J  inch  in  diameter  ? 


Fio.  1 


581. 

Observe  that  Fig.  ABCD  is  a  parallelogram. 

Its  altitude  is  what  of  the  circle? 

Its  base  is  what  of  the  circle? 

The  triangles  of  the  circle  are  what  part  of 
the  parallelogram  ? 

•  How  may  we  find  the  area  of  the  parallelo- 
gram ?     Of  the  circle  ? 


Fifi.  2 


Summary 

The  area  of  a  circle  is  equal  to  one  half  of  the  product  of  its 
circumference  hy  its  radius. 

By  geometry  it  is  proved  also  that  the  area  of  a 
circle  is  equal  to  .7854  of  the  square  of  its 
diameter,  or  3.1416  times  the  square  of  its  radius. 

How  may  we  find  the  area  of  a  circle  when  the  radius 
is  given  ?  when  the  diameter  is  given  ?  when  the  circum- 
ference is  given  ? 

582.     Written 

In  examples  1-12  find  the  area  of  a  circle  from  the  term,  given, 
letting  D,  R,  and  Q  stand  for  diameter,  radius,  and  circumference, 
respect 


350  GRAMMAR  SCHOOL   ARITHMETIC 

1.  D  =  40  in.  5.  C  =  9.4248  in.  9.  C  =  25.1328  ft. 

2.  D=102m.  6.  C  =  3.1416  mi.  lo.  D  =  60Km. 

3.  R  =  25  cm.  7.  D  =  124  rd.  11.  C  =  31.416  yd. 

4.  R  =  2  ft.  6  in.  .    8.  R  =  35  cm.  12.  R  =  2  mi. 

13.  A  horse  tethered  by  a  50-foot  rope  in  an  open  field  can 
graze  over  how  many  square  feet  of  land  ? 

14.  A  cow  is  tied  by  a  rope  100  ft.  long  at  the  corner  of  a 
rectangular  pasture  inclosed  by  a  fence. 

a.  What  part  of  an  acre  of  ground  can  she  graze  over  ? 

h.  If  she  is  tied  to  the  fence  at  the  middle  of  one  side  of 
the  pasture,  how  much  land  can  she  graze  over,  the  pasture 
being  more  than  200  ft.  long  and  wide  ? 

15.  On  a  city  map  the  center  of  the  city  is  indicated  by  a 
dot,  and  a  circle  is  drawn  to  include  all  that  part  which  is  not 
more  than  half  a  mile  from  the  center,  another  to  include  all 
that  is  not  more  than  a  mile  from  the  center,  and  so  on. 

a.  What  part  of  a  square  mile  is  inclosed  by  the  half-mile 
circle  ? 

h.    How  many  square  miles  are  inclosed  by  the  2-mile  circle? 

c.  By  the  mile  circle  ?  /.    By  the  2J-mile  circle  ? 

d.  By  the  3-mile  circle  ?         g.    By  the  4-mile  circle  ? 

e.  By  the  1  J-mile  circle  ? 

16.  If  i>2  =  841,  a,  what  is  i>  ?     h.    What  is  C? 

17.  If  i>2=  225,  a.  what  is  (7?     h.    What  is  ^? 

18.  What  is  the  diameter  of  a  circle  whose  area  is  63.6174 
sq.  ft.  ?      Statement  of  Relation :  .7854  x  D2  =  63.6174. 

19.  Find  the  circumference  of  a  circle  whose  area  is  12.5664 
square  meters. 

20.  Find  in  meters  the  radius  of  a  circle  whose  area  is 
38.4846  square  decimeters. 


SOLIDS  351 


SOLIDS 


Note.  —  In  the  study  of  solid  figures  a  full  set  of  models  should  be  in  con- 
stant use. 

583.  A  solid  is  anything  that  has  lengthy  breadth^  and  thickness. 

Anything  that  occupies  space  is  a  solid.  Any  portion  of  space  may  be 
considered  as  a  solid. 

A  solid  figure  is  bounded  by  surfaces.    By  what  are  plane  figures  bounded  ? 

584.  The  side^  or  face,  on  which  a  solid  may  he  supposed  to  rest 
is  called  its  base. 

STUDY  OF  PRISMS 

585.  A  solid  having  two  bases  which  are  equal  parallel  poly- 
gons., and  whose  other  sides  are  parallelograms.,  is  a  prism. 

586.  Prisms  are  named  according  to  the  number  of  sides 
which  their  bases  have,  as  triangular,  quadrangular,  pentagonal, 
hexagonal,  etc. 

587.  A  prism  whose  bases  and  other  faces  are  rectangles  is  a 
rectangular  prism. 

588.  A  prism  whose  bases  are  squares  and  whose  other  faces 
are  equal  rectangles  is  a  square  prism. 

How  may  the  surface  of  any  prism  be  found  ? 

589.  The  volume  of  a  rectangular  prism  is  equal  to  the  product 
of  its  three  dimensions. 

590.  The  volume  of  any  prism  is  equal  to  the  area  of  the  base., 
multiplied  by  the  altitude. 

591.  Written 

1.  Find  the  entire  surface  of  a  prism  whose  bases  are  squares 
13  inches  on  a  side  and  whose  altitude  is  2  feet. 


352  GRAMMAR  SCHOOL   ARITHMETIC 

2.  Find  the  entire  surface  of  a  rectangular  prism  whose 
dimensions  are  30  in.,  3  ft.,  and  4  ft.  6  in. 

3.  Find  the  contents  of  a  prism  whose  base  is  6  ft.  square 
and  whose  altitude  is  90  in. 

4.  What  is  the  volume  of  a  rectangular  prism  whose  dimen- 
sions are  2  ft.,  1  ft.  6  in.,  and  38  in.? 

— n -|      5.    What  is  the  volume  of  a  hexagonal  prism  the 

V' JM    ^^^^  ^^  whose  base  is  748  square  inches  and  whose 
I         altitude  is  3  feet? 

6.    a.  Find   the   entire   surface  of   the  prism  in 
in  Fig.  1. 

h.    Find  the  volume  of  the  prism  in  Fig.  1. 
^1^      7.    The  volume  of  a  square  prism  is  7776  cu.  cm. 
\Jr        Its  altitude  is  .24  m.     Find  the  length  and  breadth 
F^«- 1      of  its  base. 
Let  X  =  side  of  the  base. 

STUDY  OF   THE   CYLINDER 

Note.  —  This  treatment  is  intended  to  apply  to  the  right  circular  cylinder 
only. 

592.  A  cylinder  is  a  solid  having  two  equal  parallel  circular 
bases  and  a  convex  surface,  all  points  of  which  are  equally  distant 
from  a  straight  line  joining  the  centers  of  the  bases;  e.g.  a  round 
lead  pencil ;  a  gas  or  water  pipe ;  a  music  roll ;  a  curtain  rod. 

593.  Bring  a  cylindrical  tin  box  to  school.  Cut  a  piece  of  paper  that  will 
exactly  fit  the  convex  surface  of  the  box.  What  kind  of  figure  is  the  paper? 
Its  length  is  what  of  the  cylinder?    Its  width?    Its  area? 

Summary 

The  convex  surface  of  a  cylinder  is  the  product  of  its  altitude  and 
circumference. 


STUDY   OF   THE   CYLINDER 


353 


594.  Written 

What  is  the  convex  surface : 

1.  Of  a  cylinder  whose  circumference  is  47  in.  and  whose 
altitude  is  2  ft.  ? 

2.  Of  a  cylinder  whose  altitude  is  10  ft.  and  whose  diameter 
is  10  in.  ? 

3.  Of  a  cylinder  whose  altitude  is  20  ft.  and  whose  radius 
is  1  ft.? 

4.  Of  a  cylinder  whose  altitude  is  1  ft.  and  whose  radius 
is  2  ft.  6  in.? 

595.  Review  Figs.  1  and  2,  p.  349. 

In  the  above  figure  observe  that  the  entire  surface  of  a  cylin- 
der is  equal  to  the  convex  surface,  plus  the  sum  of  the  surfaces 
of  the  two  bases,  or  to  the  area  of  the  rectangle  ABQD. 
How  may  the  area  of  the  rectangle  A  BCD  be  found? 


Convex  Surface 


Circiiviference 


Alt.   )Alt.-\-B 


Summary 

The  entire  surface  of  a  cylinder  is  equal  to  the  product  of  the 
circumference  by  the  sum  of  the  altitude  and  radius. 

596.     Written 

Find  the  entire  surface  of  a  cylinder : 

1.    Whose  diameter  and  altitude  are  3  ft.  and  50  ft. 


354 


GRAMMAR  SCHOOL  ARITHMETIC 


2.  Whose  radius  and  altitude  are  1  ft.  and  10  ft. 

3.  Whose  circumference  and  altitude  are  respectively  25.1328 
in.  and  12  in. 

4.  Whose  circumference  and  altitude  are  31.416  meters  and 
20,000  millimeters. 

5.  Whose  diameter  and  altitude  are  20  in.  and  20  in. 

597.   Observe  that  a  cylinder  (Fig.  1)  may  be  divided  into  any  number  of 


Fig.  1  Fig.  'I 

equal  sections  (Fig.  2),  each  of  which  is  approximately  a  triangular  prism. 

The  volume  of  all  of  these  sections  com- 
bined is  equal  to  one  half  that  of  a 
rectangular  prism  (Fig.  3)  whose  di- 
mensions are  the  circumference,  alti- 
tude, and  radius,  respectively,  of  the 
cylinder. 

How  may  we  find  the  volume  of  the 
rectangular  prism?  of  the  cylinder? 


Circumference 

FiQ.  8 


Summary 

The  volume  of  a  cylinder  is  equal  to  one  half  of  the  product  of 
its  circumference^  altitude^  and  radius. 

One  half  of  the  product  of  the  circumference  and  radius  =  what?  In 
w^hat  other  form,  then,  may  the  above  summary  be  stated  ? 

598.     Written 

1.  Find  the  volumes  of  cylinders^  having  given  dimensions  as 
follows : 

a.  Alt.  8  in.,  D.  5  in.  d.  Alt.  7  ft.  2  in.,  D.  9  in." 

b.  Alt.  3  ft.,  D.  2  ft.  e.    Alt.  30  ft.,  D.  20  in. 

(?.    Alt.  1  m.,  R.  4  dm.  /.    Alt.  25  dm.,  cir.  37.6992  m. 


STUDY   OF   THE   CONE  355 

g.   Alt.  1  ft.,  cir.  3.1416  yd.        i.    Alt.  10  ft.,  cir.  7.854  in. 
h.    Alt.  80  ft.,   cir.  49.912  ft.     j,    R.  85  cm.,  alt.  5  m. 

2.  How  many  gallons  of  water  will  fill  a  cylindrical  pail 
11  in.  deep  and  9  in.  in  diameter  ?     (Indicate  the  work  first.) 

3.  The  reservoir  of  my  student  lamp  is  a  cylinder  7  in. 
high  and  3|  in.  in  diameter.  How  much  more  or  less  than  a 
quart  of  oil  will  it  hold  ? 

4.  A  cylindrical  cistern  is  6  ft.  in  diameter  and  7  ft.  deep. 
How  many  barrels  of  water  will  it  hold  ?     (Indicate  the  work  first.) 

5.  How  many  cubic  feet  of  compressed  gas  can  be  stored 
in  a  steel  cylinder  4  ft.  long  and  9  in.  in  diameter  ? 

6.  How  many  cubic  feet  of  wood  are  there  in  a  log  of  uni- 
form diameter,  whose  circumference  is  7.854  ft.  and  whose 
length- is  18  ft.? 

7.  A  farmer  has  a  cylindrical  silo  12  ft.  in  diameter  and 
30  ft.  high.  How  many  cubic  feet  of  ensilage  can  he  store 
in  it  ? 

8.  How  many  cubic  feet  of  iron,  are  there  in  an  iron  wire 
10,000  ft.  long  and  |  of  an  inch  in  diameter  ? 

9.  On  the  roof  of  Mr.  Gowing's  cottage  is  a  cylindrical 
water  tank  into  which  water  is  pumped  from  the  lake  below. 
It  is  5J  ft.  deep  and  3|-  ft.  in  diameter. 

a.    How  many  gallons  of  water  will  it  hold  ? 
h.    How  deep  is  the  water  in  the  tank  when  it  contains  100 
gallons  ?      (L®^  x  —  the  depth,  and  form  an  equation.) 

10.    Make  and  solve  five  problems  about  cylinders. 

STUDY  OF   THE   CONE 

599.    A  cone  is  a  solid  whose  base  is  a  circle^  and  whose  convex 
surface  tapers  uniformly  to  a  point  called  the  vertex. 


356 


GRAMMAR  SCHOOL  ARITHMETIC 


600.  The  altitude  of  a  cone  is  the  perpendicular 
distance  from  the  vertex  to  the  center  of  the  base. 

601.  The  slant  height  of  a  cone  is  the  distance 
from  the  vertex  to  any  point  in  the  circumference  of 
the  base. 


602.  The  convex  surface  of  a  cone  may  be  considered  as 
made  up  of  any  number  of  equal  triangles,  each  triangle  having  for  its 
altitude  the  slant  height  of  the  cone,  and  for  its  base  one  of  the  equal 
parts  of  the  circumference  of  the 
base  of  the  cone. 

The  sum  of  the  bases  of  the 
triangles  is  what  of  the  base  of 
the  cone?  The  triangles  that 
form  the  convex  surface  of  the 
cone  are  equal  to  what  part  of  the 
area  of  the  rectangle  ABCD? 

The  base  of  the  rectangle  (Fig. 
2)  is  what  of  the  cone  (Fig.  1)  ? 

The  altitude  of  the  rectangle  is  what  of  the  cone? 

The  area  of  the  rectangle  is  found  how  ? 

The  convex  surface  of  the  cone  is  what  part  of  the  area  of  the  rectangle? 

Summary 
7^e  convex  surface  of  a  cone  is  equal  to  one  half  of  the  product 
of  the  circumference  of  the  base  by  the  slant  height. 

603.  By  section  581  the  area  of  the  base  of  a  cone  is  equal  to  one 
half  of  the  area  of  a  rectangle  whose  dimensions  are  the  circum- 
ference and  radius  of  the  base  of  the  cone. 

Show  this  by  a  drawing. 

604.  Adding  this  to  the  convex  surface,  the  entire  surface  of  a 
cone  is  equal  to  one  half  of  the  product  of  the  circumference  by  the 
sum  of  the  slant  height  and  the  radius  of  the  base. 

This  rnay  be  shown  by  adding  to  the  rectangle,  Fig.  2,  a  rectangle  of 
equal  length,  with  an  altitude  equal  to  the  radius  of  the  base  of  the  cone. 


STUDY  OF  THE   CONE  357 

605.  Written 

1.  The  altitude  of  a  conical  spire  is  12  ft.  Its  base  is  10  ft. 
in  diameter.  Find  (^x)  its  slant  height ;  (by  its  convex  sur- 
face ;   (c)  its  entire  surface. 

2.  The  circumference  of  the  base  of  a  cone  is  188.496  in. 
The  slant  height  is  6  ft.  6  in.  Find  (a)  the  radius  of  the  base 
of  the  cone ;  (5)  the  altitude  of  the  cone ;  (c)  the  convex 
surface  of  the  cone ;   (^d)  the  entire  surface  of  the  cone. 

3.  Make  and  solve  other  problems  on  the  cone. 

606.  It  is  proved  by  geometry  that  the  volume 
of  a  cone  is  equal  to  one  third  of  the  volume  of 
a  cylinder  of  the  same  base  and  altitude. 

This  may  be  verified  by  filling  a  hollow  tin  cone  with 
water  and  pouring  it  into  a  cylinder  of  the  same  base 
and  altitude.  When  the  cone  has  been  emptied  once, 
the  depth  of  the  water  in  the  cylinder  is  what  part  of 
the  height  of  the  cylinder? 

607.  It  follows  from  the  above  statement  that  the  volume  of  a 
cone  may  be  found  by  taking  one  sixth  of  the  product  of  its  altitude, 
the  circumference  of  its  base,  arid  the  radius  of  its  base  ;  or,  by 
multiplying  the  area  of  its  base  by  one  third  of  its  altitude, 

608.  Written 

1.  Find  the  volume  of  a  cone  whose  altitude  is  42  in.  and 
the  area  of  whose  base  is  7  sq.  ft. 

2.  What  is  the  volume  of  a  cone  the  radius  of  whose  base  is 
20  in.  and  whose  altitude  is  3  ft.  ? 

3.  Find  the  volume  of  a  cone  whose  circumference  is  219.912 
centimeters  and  whose  altitude  is  1  meter. 

4.  Make  and  solve  other  problems  on  the  cone,  using  dimen- 
sions given  for  cylinders  in  section  598. 


358 


GRAMMAR  SCHOOL  ARITHMETIC 


STUDY  OF  REGULAR  PYRAMIDS 

609.    A  regular  pyramid  is  a  solid  whose  base  is  a  regular 
polygon^  and  whose  other  faces  are  equal  triangles  meeting  at  a 

point  called  the  vertex. 

610.  Pyramids  are 
named  from  their  bases, 
as  triangular  pyramids, 
square  pyramids,  hex- 
agonal pyramids,  etc. 

611.  The  altitude  of  a  regular  pyramid  is  the  distance  from  its 
vertex  to  the  middle  of  its  base. 

612.  The  slant  height  of  a  regular  pyramid  is  the  altitude  of 
one  of  its  triangular  faces. 

613.  The  lateral  surface  of  a  pyramid  is  the  combined  surface 
of  all  of  its  triangular  faces. 

614.  How  may  the  surface  of  each  triangular  face  be  found  ? 
Of  all  of  them  ?     How  may  the  entire  surface  be  found? 


Summary 

The  lateral  surface  of  a  regular  pyramid  is  the  product  of  the 
perimeter  of  its  base  by  one  half  of  its  slant  height.      The  entire 
surface  is  the  sum  of  the  lateral  surface  and 
the  base. 

615.  It  is  proved  by  geometry  that  the 
volume  of  a  regular  pyramid  is  equal  to  one 
third  of  the  volume  of  a  regular  prism  having 
the  same  base  and  altitude. 

How  is  the  volume  of  a  prism  found?  Of  a 
pyramid  having  the  same  base  and  altitude  as  the 
prism  ? 


STUDY  OF  THE  SPHERE 


359 


616.  Written 

1.  What  is  the  lateral  surface  of  a  regular  triangular  pyramid 
whose  slant  height  is  25  feet  and  one  side  of  whose  base  is  15 
feet? 

2.  The  roof  of  a  tower  is  in  the  form  of  a  pyramid  whose 
base  is  9  ft.  square  and  whose  altitude  is  6  ft. 

a.  Find  its  slant  height. 

b.  Find  its  lateral  surface. 

c.  Find  its  volume. 

3.  a.  Find  the  entire  surface  of  a  drawing  model  in  the 
form  of  a  square  pyramid  whose  altitude  is  6  in.  and  the  side 
of  whose  base  is  4  in. 

b.    Find  its  volume. 

4.  Find  the  lateral  surface  of  an  octagonal  church  spire 
each  side  of  whose  base  is  5  ft.  and  whose  slant  height  is 
40  ft. 

5.  Find  the  volume  of  a  pyramid  whose  altitude  is  16.5 
m.  and  the  area  of  whose  base  is  170,000  sq.  cm. 

STUDY  OF  THE   SPHERE 

617.  A  sphere  is  a  solid  bounded  by  a  surface^  every  point  in 
which  is  equally  distant  from  a  point  within 
called  the  center. 


618.  The  diameter  of  a  sphere  is  a  straight 
line  passing  through  its  center  and  terminating 
in  its  surface. 

619.  The  radius  of  a  sphere  is  a  line  drawn  from  its  center  to 
any  point  in  its  surface. 

620.  The  circumference  of  a  sphere  is  the  circumference  of  a 
circle  whose  radius  and  center  are  those  of  the  sphere. 


360  GRAMMAR  SCHOOL   ARITHMETIC 

621.  It  is  proved  by  geometry  that  the  surface  of  a  sphere  u 
the  product  of  its  diameter  and  circumference. 

This  is  the  same  as  the  square  of  the  diameter  multiplied  by  3.1416. 
Explain. 

It  is  also  four  times  the  square  of  the  radius  multiplied  by  3.1416. 
Explain. 

622.  Written 

1.  Find  the  surface  of  a  sphere  whose  diameter  is  100  ft. 

2.  Assuming  the  earth  to  be  a  sphere  (it  is  nearly  a  sphere) 
and  its  radius  to  be  4000  miles,  what  is  its  area  ? 

3.  What  is  the  surface  of  a  sphere  whose  circumference  is 
37.6992  Km.  ? 

4.  What  is  the  surface  of  a  sphere  whose  circumference  is 
39.27  inches? 

5.  Assuming  the  diameter  of  the  moon  to  be  2150  miles, 
what  is  its  area  ? 

623.  A  sphere  may  be  supposed  to  be  made  up  of  a  number 
of  pyramids,  as  shown  in  the  cut. 

By  sections  590  and  615,  the  volume  of  each  of 
these  pyramids  is  equal  to  the  area  of  the  base 
multiplied  by  one  third  of  the  altitude.  How 
does  the  sum  of  the  bases  of  the  pyramids  com- 
pare with  the  area  of  the  sphere  ?  The  altitude  of 
each  pyramid  is  what  of  the  sphere?  How,  then, 
may  we  find  the  contents  of  a  sphere? 

Summary 

The  volume  of  a  sphere  is  equal  to  one  third  of  the  product  of 
its  radius  and  area. 

This  is  the  same  as  f  of  the  cube  of  the  radius  multiplied  by  3.1416. 
Explain. 


SIMILAR  SURFACES  361 

624.  Written 

1.  Find  the  volume  of  a  sphere  :  a.  Whose  diameter  is  200 
feet.  b.  Whose  radius  is  100  meters,  c.  Whose  circumfer- 
ence is  157.08  cm.  d.  Whose  circumference  is  78.54  inches. 
e.   Whose  radius  is  1  foot. 

2.  Assuming  the  diameter  of  the  moon  to  be  2000  miles  (it 
is  nearly  2150  miles),  what  is  its  volume  ? 

3.  Assuming  the  diameter  of  the  earth  to  be  8000  miles, 
what  is  its  volume  ? 

4.  a.  Find  the  volume  of  a  cone  whose  altitude  and  the 
diameter  of  whose  base  are  each  20  inches. 

h.  Using  the  answer  to  question  a,  find  the  volume  of  a 
cylinder  having  the  same  dimensions. 

c.  Find  the  volume  of  a  sphere  whose  diameter  is  20  inches. 

d.  The  volume  of  the  sphere  is  how  many  times  that  of  the 
cone  ? 

e.  The  volume  of  the  cylinder  is  how  many  times  that  of  the 
cone  ?  of  the  sphere  ? 

/.  Find  the  volume  of  another  cone,  sphere,  and  cylinder 
whose  diameters  and  altitudes  are  all  equal.  Compare  them. 
Geometry  proves  that  this  relation  always  exists. 

SIMILAR  SURFACES 

Note.  —  Review  proportion. 

625.  Figures  that  have  the  same  shape^  though  they  may  differ 
in  size,  are  similar  ;  e.g.  all  circles  are  similar  ;  all  regular  poly- 


O 
Similar  Figures  Similar  Figures 


362  GRAMMAR  SCHOOL   ARITHMETIC 

gons  of  the  same  number  of  sides  are  similar  ;  two  rectangles 
are  similar  if  the  length  and  breadth  of  each  have  the  same  ratio. 

626.  It  is  proved  by  geometry  that  if  two  figures  are  similar^ 
any  two  lines  of  one  figure  have  the  same  ratio  as  the  correspond- 
ing two  lines  of  the  other  figure  ;  and  a  line  of  one  figure  has  the 
same  ratio  to  the  corresponding  line  of  the  other  figure  that  any 
other  line  of  the  first  figure  has  to  the  corresponding  line  of  the 
other  figure. 

For  example,  in  the  figures  shown  in  section  625, 

AB:AC=:  A'B'  lA'C  EF :  FG  =  E' F'  :F'G' 

AB  :  A'B'  =BC:B'C'  DG  :  D'G'  =  FG :  EG' 

If  the  side  AB  equals  21  ft.,  the  side  .4  C  12  ft.,  and  the  side  A'B'  14  ft.,  we 
may  find  the  length  of  the  side  A'C  by  the  following  proportion : 

21  ft.  :12  ft.  =  14  ft.:  a;  ft. 
Find  the  value  oi  x. 

627.  Written 

1.    In  Figs.  1,  2,  3,  and  4  : 

a.  If  AB  =  14:  ft.,  A'B'=2S  ft.,  and  BO  =11  ft.,  what  is 
the  length  of  ^^(7' ? 

b.  If  JEF=  15  rd.,  Fa  =  10  rd.,  and  F'F'  =  18  rd.,  what  is 
the  length  of  i^' 6^'? 

c.  li  Da  =  21  mi.,  B' a'  =  33  mi.,  and  FG  =  18  mi.,  what 

is  the  value  of  F'  G'  ? 
^'''''///7^{/,^';y^^  2.  A  man,  desiring  to  know  the 
height  of  a  tree  which  stood  on  level 
ground,  drove  a  stick  into  the  earth 
in  a  vertical  position,  and  it  meas- 
ured 3  ft.  above  ground.  Its 
'W^  "  shadow   measured  45   in.      At  the 

same  moment  the  tree  cast  a  shadow  39  ft.  long.     How  tall 
was  the  tree  ? 


SIMILAR  SURFACES 


363 


3.  A  rectangular  field  is  70  rd.  long  and  50  rd.  wide;  what 
is  the  length  of  a  similar  field  whose  width  is  12J  rd.  ? 

4.  One  side  and  the  diagonal  of  a  quadrilateral  are  respec- 
tively 18  ft.  and  44  ft.  Find  the  corresponding  side  of  a  similar 
quadrilateral  whose  diagonal  is  110  ft. 

5.  A  boy  found  the  height  of  a  flagstaff  as  follows  : 

He  found  that  he  could  hold  a  cane  upright  just  30  in.  away 
from  his  eye.     He  placed  his  thumb  22|  in.  from  the  top  of  the 
cane,  pinned  a  card  on  the  flagstaff  just  as  high  as  his 
eye,  and  walked  backward  until  he  could  just  see 
the  paper  by  looking  across  the  top  of  his  thumb 
where  he  held  the  cane,  and  see  the  top  of        /' 
the  flagstaff  by  looking  across  the  top        ,.'''' 
of  the  cane.     He  then  found  by         /' 

measurement  that  he  stood  72  Uye'^^^^r^-^- 

ft.  from  the  flagstaff  while  tak- 


72' 


ing  the  observation,  and  that   the   card  was  5  ft.  from  the 
ground.     How  high  was  the  flagstaff  ? 

6.  Two  boys,  wishing  to  know  the 
^  width  of  a  river  and  having  no  boat, 
constructed  the  right  triangle  ABO  by 
driving  three  stakes.  They  sighted  from 
A,  across  (7,  to  the  opposite  bank,  at  JE, 
and  drove  a  stake  at  D,  so  as  to  make 
the  right  triangle  ODE.  They  then 
measured  AB,  BO,  and  01),  and  found 
DU.     How  wide  was  the  river  ? 

628.    It  is  proved  by  geometry  that  the  areas  of  similar  surfaces 
are  to  each  other  as  the  squares  of  any  two  corresponding  lines. 

Thus,  on  page  361,  if  the  side  AB  oi  Fig.  1  is  21  ft.,  the  side  A'B'  of 
Fig.  2,  14  ft.,  and  the  area  of  Fig.  1,  96  sq.  ft.,  we  may  find  the  area  of 


364  GRAMMAR  SCHOOL  ARITHMETIC 

Fig.  2  by  making  the  proportion  ' 

212:142  =  96:a;. 
2        2      32 

«^^-"^'  ^-'-^tft^'  =  f  =  42t  sq.ft.     Ans. 

3        3 
If  the  area  of  Fig.  3  is  48  sq.  ft.  and  of  Fig.  4,  120  sq.  ft.,  and  the  side 
DE  of  Fig.  3  is  6  ft.,  the  side  D'E'  of  Fig.  4  may  be  found  by  making  the 
proportion, 

48:120  =  62:x2. 

15 
Solving,  a;2  =  M2<A>if  =  90. 


Since  a;2  =  90, 

a:  =  V90,  or  9.48+  f t.    Ans, 
629.     Written 

1.  The  side  of  a  triangle  is  7  inches  and  its  area  23  square 
inches.  The  corresponding  side  of  a  similar  triangle  is  10| 
inches.     Find  its  area. 

2.  The  corresponding  sides  of  two  similar  rectangles  are 
19  rods  and  152  rods.  The  area  of  the  second  is  5670  square 
rods.     What  is  the  area  of  the  first  ? 

3.  A  circle  is  4  inches  in  diameter  ;  another  is  8  inches  in 
diameter.     What  is  the  ratio  of  their  areas  ? 

4.  A  circle  has  an  area  of  16  square  feet  ;  another  has  an 
area  of  64  square  feet.     What  is  the  ratio  of  their  diameters  ? 

5.  The  area  of  a  rectangle  12  feet  long  is  84  square  feet. 
What  is  the  area  of  a  similar  rectangle  6  feet  long  ? 

6.  Two  similar  fields  have  areas  of  12  acres  and  8  acres 
respectively  ;  the  larger  is  32  rods  wide.  How  wide  is  the 
smaller  ? 


LONGITUDE   AND   TIME  365 

7.  The  altitudes  of  two  similar  triangles  are  20  ft.  and 
10  ft. ;  the  area  of  the  smaller  is  80  sq.  ft.  What  is  the  area 
of  the  larger  ? 

8.  The  areas  of  two  similar  rectangles  are  8  acres  and  72 
acres  respectively.  The  diagonal  of  the  first  is  51  rods.  What 
is  the  diagonal  of  the  second  ? 

9.  An  oval  mirror  is  32  inches  long  and  has  an  area  of  600 
square  inches.  What  is  the  area  of  a  similar  mirror  whose 
length  is  40  inches  ? 

10.  Make  a  problem  to  find  the  area  of  one  of  two  similar 
figures. 

LONGITUDE   AND  TIME 

.630.  A  meridian  is  an  imaginary  line  extending  directly  north 
and  souths  on  the  surface  of  the  earthy  from  pole  to  pole.  It  is  a 
semi-circumference  of  the  earth. 

631.  A  prime  meridian  is  a  meridian  taken  as  a  starting  place 
for  the  measurement  of  distances  east  and  west  so  as  to  determine 
the  location  of  places  on  the  earth's  surface. 

By  common  consent,  the  meridian  passing  through  the  Royal  Observa- 
tory at  Greenwich,  Eng.,  is  generally  taken  as  the  prime  meridian. 

632.  Distance  east  or  west  from  the  prime  meridian^  measured 
in  degrees^  minutes,  and  seconds  is  longitude. 

Degrees,  minutes,  and  seconds  west  of  the  prime  meridian  are 
called  west  longitude  ;  east  of  the  prime  meridian  east  longitude. 

Longitude  is  measured  by  arc  measure.  Why?  The  number  of  merid- 
ians that  may  be  represented  on  a  globe  or  map  is  unlimited.  Every 
place  on  the  face  of  the  globe  may  be  supposed  to  have  its  meridian.  But 
all  places  which  lie  on  the  same  meridian  have  the  same  longitude  although 
they  may  be  thousands  of  miles  apart.  For  example,  Boston,  Mass.,  and 
Santiago,  Chile,  have  nearly  the  same  longitude,  though  widely  separated. 


GRAMMAR  SCHOOL  ARITHMETIC 


Lay  your  book  on  the  desk,  and  imagine  that  the  sun  is  in  the  ceiling 
directly  above  the  middle  of  this  drawing  of  a  hemisphere.  The  drawing 
shows  the  half  of  the  earth's  surface  that  the  sun  shines  upon.  The  other 
half  is  dark.  If  it  is  the  21st  of  March  or  September,  it  is  now  sunset  at 
the  prime  meridian,  noon  at  the  meridian  of  90°  west  longitude,  and  sunrise 
at  the  meridian  of  180°  west  longitude. 


Horth  Pole 


South  Pole 


The  earth  makes  one  rotation  toward  the  east  in  24  hours.  During  one 
rotation  all  the  meridians  will  pass  under  the  sun,  on  to  sunset,  midnight, 
sunrise,  and  noon,  finally  reaching  the  same  position  that  they  now  have. 
Every  place  on  the  earth's  surface  has  passed  under  the  sun,  and  360°  of 
longitude  have  passed  under  the  sun.  Therefore  the  number  of  degrees 
of  longitude  passing  under  the  sun  in  one  hour  is  360  h-  24,  or  15°. 

Imagine  this  drawing  to  be  a  sphere  rotating  toward  the  east.  The  sun 
remains  overhead ;  therefore  the  numbers  representing  the  hours  of  the  day 
remain  fixed,  and  the  meridians  pass  under  them. 


LONGITUDE  AND  TIME  367 

Greenwich  and  all  places  on  its  meridian  pass  into  night.  In  one  hour 
the  15°  meridian  will  be  at  six  o'clock,  the  105°  meridian  at  noon,  and  so  on. 

In  six  hours  the  90°  meridian  will  be  just  passing  the  six  o'clock  mark, 
the  180°  meridian  will  be  at  noon,  and  Greenwich  will  be  directly  opposite, 
at  midnight. 

In  twelve  hours  the  180°  meridian  will  have  passed  entirely  across  to 
6  P.M.,  and  the  meridian  of  Greenwich  will  be  just  coming  into  sight  at 
6  A.M.  The  meridians  then  in  view  will  all  be  in  east  longitude  and  will 
be  numbered  from  the  prime  meridian  at  the  left,  toward  the  right,  from 
0  to  180°  east  longitude.  That  is,  the  meridians  are  numbered  both  east 
and  west  from  the  prime  meridian  to  the  meridian  opposite,  which  is  180°. 
No  place  can  have  more  than  180°,  either  east  or  west  longitude. 

633.  The  180°  meridian,  with  slight  modifications,  has  been 
chosen  as  the  International  Date  Line.  Passing  chiefly  through 
the  Pacific  Ocean,  it  touches  no  important  body  of  land. 

Whenever  a  ship  crosses  this  line,  going  westward,  its  calendar  is  set  for- 
ward one  day ;  going  eastward,  its  calendar  is  set  back  one  day. 

634.  Oral 

Use  drawing  of  hemisphere  in  obtaining  answers. 

1.  When  it  is  noon  at  New  Orleans,  what  is  the  time  at 
Denver  ?  at  Cape  Nome  ?  at  Greenwich  ? 

2.  When  it  is  noon  at  Denver,  what  is  the  time  at  New 
Orleans?  at  Greenwich?  at  Cape  Nome? 

3.  When  it  is  noon  at  Greenwich,  what  is  the  time  at  New 
Orleans  ?  at  Denver  ?  at  Cape  Nome  ? 

4.  When  it  is  noon  at  Santiago,  what  is  the  time  at  Boston? 
at  Montevideo?  at  Rio  Janeiro? 

5.  When  it  is  noon  at  San  Francisco,  what  is  the  time  at 
Honolulu  ?  at  Charleston  ? 

6.  When  it  is  3  p.m.  at  New  York,  what  is  the  approximate 
time  at  Santiago  ?  at  Montevideo  ?  at  Rio  Janeiro  ? 


368  GRAMMAR   SCHOOL   ARITHMETIC 

7.  When  it  is  5  A.M.  at  Charleston,  what  is  the  approximate 
time  at  San  Francisco  ?  at  Honolulu  ?  at  Greenwich  ? 

8.  When  it  is  7  a.m.  at  Denver,  what  is  the  approximate 
time  at  San  Francisco  ?    at  New  York  ?  at  Greenwich  ? 

9.  The  difference  in  time  between  two  places  is  2  hr.    What 
is  the  difference  in  longitude  ? 

10.  The  difference  in  longitude  between  two  places  is  90°. 
What  is  their  difference  in  time  ? 

11.  When  it  is  9  a.m.  at  your  home,  what  is  the  time  at  a 
place  45**  farther  west  ?   at  a  place  20°  farther  east  ? 

12.  J |i| . > 

A  is  30°  west  longitude,  and  B  is  40°  east  longitude.  How 
many  degrees  of  longitude  are  there  between  the  meridian  of 
A  and  that  of  B  ? 

What  is  the  difference  in  time  between  A  and  B  ? 

635.    Written 
1.    Cape  Town  is  in  longitude  18°  28'  40'^  E.,  and  Hamburg 
is  in  longitude  9°  58'  25"  E. 

a.  What  is  their  difference  in  time  ? 

b.  When  it  is  10  a.m.  at  Cape  Town,  what  is  the  time  at 
Hamburg  ? 

c.  When  it  is  3  min.  17  sec.  before  4  A.M.  at  Hamburg,  what 
is  the  time  at  Cape  Town  ? 

a,  18°  28'  40"  The  difference  in  longitude 

9  58  25  ^^  ^^  ^^'  ^^"'    ^i^^®  *^*®  scale 

1  CN~~oo  oTw TTff  of  the  table  of  time  is  like 

■^ that  of  the  table  of  arc  meas- 

34  min.  1  sec.  Ans.  ure,  and  since  15°  of  longi- 
tude pass  under  the  sun  in  1  hr.  of  time,  15'  in  1  min.  of  time,  and  15" 
in  1  sec.  of  time,  the  number  of  hours,  minutes,  and  seconds  difference  in 


hr. 

min. 

sec. 

10 

0 
34 

0 

1 

9 

25 

59 

or 

25 

mill. 

59 

sec.  I 

hr. 

min. 

sec. 

3 

b6 
34 

43 
1 

LONGITUDE   AND   TIME  369 

time  is  -^^  as  great  as  the  number  of  degrees,  minutes,  and  seconds  difference 
in  longitude. 

Since    Hamburg    is    farther   west 
5.  10  0  0       than  Cape  Town,  its  time  is  earlier 

than  the  time  at  Cape  Town. 

t  4  A.M.     Ans. 

The   time  at  Cape  Town  is  later 
e,  3  bi5  43       than  the  time  at  Hamburg,    Why  ? 

34  1 

4  30  44 

or  30  min.  44  sec.  past  4  a.m.     Ans. 

2.  When  it  is  31  min.  30J  sec.  past  1  p.m.  at  Washington, 
D.C.,  it  is  half  past  10  a.m.  at  San  Francisco.  What  is  the 
longitude  of  Washington,  if  the  longitude  of  San  Francisco  is 
122°  25'  41"  W.  ? 

hr.  min.  sec.  The   day  begins  at  midnight. 

13  31  30|-  .  Hence,  1  p.m.  is  13  hr.  after  the 

\0  30  0  beginning  of  the  day. 

For  reasons  given  in  example 
1,  the  number  of  degrees,  min- 
utes, and  seconds  difference  in 
longitude  is  15  times  as  great  as 
the  number  of  hours,  minutes, 
and  seconds  difference  in  time, 
or  45°  22' 35". 

77°  3  6'    W.L.      Ans.  s^^^e  Washington    has    later 

time  than  San  Francisco,  it  must  be  farther  east,  therefore  nearer  the  prime 
meridian.  Hence,  it  has  a  less  longitude.  122°  25'  41"  minus  45°  22'  35" 
is  77°  3'  6". 

3.  Rome  is  12°  27'  14^'  E.L.  and  Philadelphia  75"  9'  45"  W.L. 
What  is  their  difference  in  longitude  ?    (See  Ex.  12,  p.  368.) 

Since  Philadelphia  and  Rome  are  on  opposite  sides  of  the  prime  merid* 
ian,  their  difference  in  longitude  is  the  sum  of  their  longitudes. 


3 

1 

301  Diff.  in  Time. 
15 

45° 

22' 

35"  DifT.  in  Long. 

122° 

25' 

41"  W.L. 

45 

22 

35 

370 


GRAMMAR  SCHOOL   ARITHMETIC 


In  examples  4-27  the  number  given  is  either  difference  in 
time  or  difference  in  longitude  between  two  places.  In  every 
case  find  the  one  not  given. 


4.  5  hr.  1  min.  17  sec. 

5.  1  hr.  18  min.  44  sec. 

6.  37  min.  20  sec. 

7.  2  hr.  48  sec. 

8.  8  hr.  21  min. 

9.  47°  19'  30" 

10.  18°  41' 

11.  9°  45" 

12.  12°  7'  30" 

13.  58'  15" 

14.  113°  30'  10" 

15.  107°  1'  40" 


16.  8  hr.  7  min.  22^  sec. 

17.  1  hr.  1  min.  49  sec. 

18.  160°  14'  50" 

19.  28°  40' 

20.  10  hr.  14  min.  27  sec. 

21.  46°  18' 

22.  1  hr.  2  min.  14|  sec. 

23.  48' 15" 

24.  1'49" 

25.  7  hr.  50  sec. 

26.  42°  19' 5" 

27.  170°  55' 


28.  One  place  is  in  68°  W.L.  and  another  in  53°  15'  W.L. 
What  is  their  difference  in  time  ? 

29.  Two  places  are  in  120°  47'  and  13°  50'  east  longitude 
respectively.     What  is  their  difference  in  time  ? 

30.  One  place  is  in  83°  5'  west  longitude  and   another   in 
7°  16'  15"  east  longitude.     What  is  their  difference  in  time? 

31.  It  is  12  o'clock,  midnight,  at  a  certain  place. 

a.    What  is  the  time  at  a  place  12°  15'  farther  east  ? 
h.    What  is  the  time  at  a  place  47°  18'  farther  west  ? 

32.  When  it  is  2  p.m.  at  Paris,  2°  20'  15"  E.L., 

a.    What  is  the  time  at  Melbourne,  144°  57'  45"  E.L.  ? 
I,    What  is  the  time  at  Albany,  73°  44'  45"  W.L.  ? 

33.  What  is  the  time  at  Cincinnati,  84°  26'  W.L.,  when  it  is 
11.50  A.M.  at  St.  Louis,  90°  15'  15"  W.L.  ? 


STANDARD   TIME  371 

34.  If  I  sail  from  Philadelphia,  75°  9'  45''  W.L.,  with  my 
watch  set  at  the  exact  local  time,  and,  after  sailing  a  certain 
distance,  find  that  my  watch  is  1  hr.  28  min.  40  sec.  slower 
than  the  exact  local  time  at  that  place,  assuming  that  my 
watch  has  kept  perfect  time,  what  longitude  has  the  ship 
reached  ? 

35.  A  horse  trotted  a  mile  in  2  min.  15  sec. 

a.  During  that  time,  the  race  track,  on  which  the  horse  was 
traveling,  moved  how  many  minutes  and  seconds  in  its  rotation 
about  the  earth's  axis  ? 

5.  Estimating  a  degree  of  longitude  at  that  place  to  be  equal 
to  50  miles,  how  many  miles  did  the  race  track  move  while  the 
horse  was  trotting  a  mile  ? 

36.  a,  A  railroad  train  moving  at  the  rate  of  24  miles  an 
hour,  including  stops,  travels  how  far  in  a  day  ? 

h.  The  track  on  which  the  train  runs  moves  how  many  miles 
a  day,  assuming  a  degree  of  longitude  at  that  latitude  to  be 
50  miles  ? 

STANDARD  TIME 

636.  The  railroad  companies  of  this  country  and  Canada 
have  agreed  upon  a  division  of  the  country  into  four  time  belts, 
extending  north  and  south.  All  places  in  each  belt  take  the 
time  of  the  meridian  which  passes  through  or  near  the  middle 
of  the  belt.  This  time  is  called  standard  time.  The  belts  are 
as  follows :  Eastern^  Central^  Mountain^  and  Pacific. 

A  similar  system  of  standard  time  is  used  in  other  parts  of 
the  world. 

The  standard  meridian  for  the  Eastern  belt  is  the  75th,  for 
the  Central  belt  the  90th,  for  the  Mountain  belt  the  105th,  and 
for  the  Pacific  belt  the  120th, 


372 


GRAMMAR   SCHOOL   ARITHMETIC 


These  standard  meridians  are  15  degrees  apart:  when  it  is 
noon  in  the  Eastern  belt,  it  is  11  A.M.  in  the  Central  belt,  10  A.M. 
in  the  Mountain  belt,  and  9  a.m.  in  the  Pacific  belt. 

PACIFIC  TIME +•'•120-°  MOUNTAIN  TIME  +105°  CENTRAL  TIME  +90°  EASTERN  TIME  +  75° 


In  going  westward  from  one  time  belt  into  another,  the  trav- 
eler sets  his  watch  back  one  hour.  In  traveling  eastward  he 
sets  his  watch  ahead  one  hour. 

When  it  is  noon  on  the  standard  meridian  of  a  time  belt,  it 
is  called  noon  at  all  places  in  the  belt. 

Standard  time  is  not  the  true  solar  or  local  time,  except  for  places  situ- 
ated on  the  standard  meridians.  Yet  it  can  vary  but  little  more  than 
thirty  minutes  from  the  true  time,  and  its  uniformity  is  a  convenience. 

Standard  time  is  used  not  only  by  the  railroads,  but  also  by  people  gen- 
erally. The  exact  time  is  telegraphed  daily  to  all  sections  of  the  country 
from  the  Naval  Observatory  at  Washington. 

637.    Oral 

1.  When  it  is  5  P.M.  Mountain  time,  what  is  the  time  in  the 
Pacific  belt  ? 


REVIEW   AND   PKACTICE  373 

2.  When  it  is  11  A.M.  Pacific  time,  what  is  the  Central  time  ? 

3.  In  traveling  from  San  Francisco  to  New  York,  how  many 
times  do  I  change  my  watch,  and  do  I  set  it  ahead  or  back  ? 

4.  When  it  is  4  A.M.  at  Augusta,  Me.,  what  is  the  standard 
time  at  St.  Louis  ? 

5.  When  it  is  1  p.m.  Mountain  time  at  Denver,  what  time  is 
it  at  Washington,  D.C.  ? 

6.  What  is  the  Pacific  time  at  San  Francisco  when  it  is 
5  P.M.  at  Chicago? 

638.  Written 

1.  What  is  the  local  time  at  Quebec,  71°  12'  15''  W.L., 
when  the  standard  time  at  that  place  is  7.30  a.m.  ? 

2.  What  is  the  difference  between  local  time  and  standard 
time  in  Chicago,  whose  longitude  is  87°  36'  42"  W.  ?  . 

3.  When  it  is  6  P.M.,  standard  time,  at  San  Francisco, 
122°  25'  41"  W.,  what  is  the  local  time  ? 

REVIEW  AND  PRACTICE 

639.  Oral 

1.  Express  in  words:  4009;  350.01259;  CXLVIII ; 
MCMX. 

2.  For  what  is  the  decimal  point  used  ? 

3.  Moving  a  figure  three  places  to  the  left  has  what  effect 
on  its  value  ?     Two  places  to  the  right  ? 

4.  Moving  the  decimal  point  two  places  to  the  right  has 
what  effect  on  the  value  of  the  number  in  which  it  is  placed  ? 
One  place  to  the  left  ? 

5.  State  three  principles  of  Roman  notation. 

6.  Describe  two  methods  of  testing  results  in  subtraction. 


374  GRAMMAR  SCHOOL  ARITHMETIC 

7.  Which  term  in  subtraction  corresponds  to  the  sum  in 
addition  ?     It  is  the  sum  of  what  ? 

8.  Which  terms  in  multiplication  are  factors? 

9.  What  is  the  shortest  way  to  multiply  an  integer  by  100  ? 
To  multiply  an  integer  by  7000  ? 

10.    3675x100=?     600x7000=?     98x100=? 

640.    Written 

In  examples  1-4,  add  a7id  test  results: 
1. 


235 

2.  8397 

3.  $18.79 

4.  i69. 

49 

Qb 

4.65 

72.35 

807 

482 

82.04 

670.48 

9063 

39 

9.00 

8359.20 

584 

910 

501.83 

2517.03 

5369 

8765 

7.62 

932.45 

70810 

1974 

9.30 

8534.06 

52479 

193 

18.49 

92.08 

1379 

8370 

43.86 

801.64 

95468 

246 

97.53 

17.32 

3007 

98 

68.12 

84.63 

88894 

4839 

835.27 

91.02 

5.  From  900,003.2  take  100.01. 

6.  Multiply  374  by  268  and  read  the  partial  products. 

7.  468,316  is  the  product  of  68,  71,  and  what  other  factor  ? 

8.  Find  the  value  of  4837  +  32  x  1800  - 1728  -^  72. 

9.  Find  the  value  of  (4837  +  32)  x  (1800  -  1728)  -*-  72. 
10.    Find  the  prime  factors  of  36,465. 

641.    Oral 
1.    How  many  acres  of  land  can  be   bought   for   $18,200, 
if  every  two  acres  cost  $182? 


REVIEW  AND  PRACTICE  375 

2.  Test  for  divisibility  by  2,  4,  3,  5,  25,  and  9,  each  of  the 
following  numbers  :  2352;  86,543,400;  793,422;  123,797. 

3.  Name  the  prime  numbers  from  1  to  100. 

4.  Name  two  composite  numbers  that  are  prime   to   each 
other. 

5.  When  is  a  fraction  in  lowest  terms  ? 

6.  When  is  a  number  in  simplest  form  ? 

7.  What  common  fraction  is  equal  to  .50?    .33J?    .121? 
.60?  .75?    .66-1?    .80?    .40?    .90? 

8.  What  is  the  shortest  way  to  multiply  an  integer  by  700  ? 

9.  Multiply  24.651  by  100  ;  by  1000. 
10.    Name  four  aliquot  parts  of  50. 

642.     Written 

1.  Which  of  the  following  numbers  are  prime  :  137  ;   361 ; 
247  ;  381  ;  215  ;  897  ? 

2.  Find  the  L.  C.  M.  of  63,  66,  and  77. 

3.  Find  two  numbers  whose  sum  is  835,  and  whose  differ- 
ence is  473. 

4.  What  is  the  greatest  common  divisor  of  396  and  468  ? 


Simplify 


13| 


6.  Simplify  if  X  If -i-(^\  +  f). 

7.  A  man  made  his  will,  giving  his  son  13210,  which  was  | 
of  his  estate  ;  to  his  daughter  -^^  of  his  estate  ;  and  the  re- 
mainder to  his  wife. 

a.    How  much  did  the  daughter  receive  ? 
h.    How  much  did  the  wife  receive  ? 

8.  When  I  of  a  yard  of  cloth  costs  1 2. 40,  how  many  yards 
can  be  bought  for  119.20  ? 


376  GRAMMAR  SCHOOL  ARITHMETIC 

9.    What  fraction  of  24 1  is  6|  ? 

10.    A  boy  spent  |  of  his  money  and  then  earned  65  cents. 
He  iuhen  had  |  of  his  original  sum.     How  much  had  he  at  first  ? 

643.     Oral 

1.  What  is  the  easiest  way  to  divide  an  integer  by  100  ? 
To  divide  a  decimal  by  1000  ? 

2.  What  is  the  easiest  way  to  divide  a  number  by  25  ? 
by  125  ? 

3.  Name  a  denominate  number  that  is  not  compound. 
Name  a  denominate  number  that  is  compound. 

4.  What  is  the  cost  of  3000  shingles  at  15.00  per  M  ? 

5.  Name  some  article  that  weighs  about  one  pound ;  about 
two  pounds  ;  about  three  pounds  ;  about  fifty  pounds. 

6.  Without  measuring,  draw  a  line  six  feet  long  on  the 
blackboard.  Draw  another  line  two  thirds  as  long.  Meas- 
ure and  correct  your  drawings. 

7.  My  watch  chain  of  14  k.  gold  is  worn  out,  and  the  jew- 
eler will  allow  me  56  j^  per  pennyweight  for  it.  If  it  weighs 
10  pwt.,  how  much  will  I  be  allowed  for  it  ?  The  value  of  the 
gold  is  in  proportion  to  its  fineness.  How  much  would  my 
chain  be  worth  if  it  were  10  k.  gold  ? 

8.  What  is  the  cost  of  10  quires  of  paper  at  the  rate  of 
80^  per  ream  ? 

9.  An  arc  of  30°  is  what  part  of  a  circumference  ? 

10.    a.   How  many  seconds  are  there  in  an  hour  ? 
h.  What  is  the  difference  in  time  between  two  places,  one  of 
which  is  15°  W.L.,  and  the  other  45°  E.L.? 


82.57 

937.48 

64.37 

9.84 
83.06 


REVIEW   AND  PRACTICE  377 

644.  Written 
1.    Add:  2.    A  man  owning  |  of  a  boat  sold  |  of  his 

1243.76  share  for  11785.     What  was  the  value  of  the 

58.19  boat  at  that  rate  ? 

23.79  3^    I  of  a  number  exceeds  |  of  the  number 

1-^4  by  4821.     What  is  the  number  ? 

4.  A  miller  bought  wheat  at  65|  ^  per  bushel 
and  sold  it  at  75|.^  per  bushel,  gaining  in  all 
$117.     How  many  bushels  did  he  buy  and  sell  ? 

5.  Factor  17,280. 

72.00  6-    What  fraction  of  a  bushel  is  3  pk.  7  qt. 

9.73  Ipt.? 

64.58  7.    What  fraction  of  a  gallon  of  water  can  be 

7.86  held  in  a  tin  box  4  in.  square  and  3  in.  deep  ? 

•98  8.    240  rd.  is  what  fraction  of  a  mile  ? 

9.    Reduce  35,816  in.  to  higher  denomina- 

28-62  tions. 
9.18 

'   ^  10.    What  is  the  cost  of  digging  a  cellar  25' 

519*08   ^^  ^^'  ^^  ^2^  ^^  ^^^  p®^  ^^^^^  y^^^  ^ 

645.  Oral 

1.  A  flagstone  is  5  ft.  long  and  3  ft.  wide.      How  thick 
must  it  be  to  contain  5  cu.  ft.  of  stone  ? 

2.  How  many  cubic  yards  are  there  in  a  block  of   stone 
27  ft.  long,  6  ft.  wide,  and  3  ft.  thick  ? 

(Think  the  problem  through  before  you  perform  any  operation.) 

3.  A  piece  of  cloth  is  36  yd.  long  and  2  ft.  wide.     How 
many  square  yards  of  cloth  does  it  contain  ? 

4.  What  is  the  length  of   one  degree  of   a  circumference 
which  measures  360  inches  ? 


378  GRAMMAR   SCHOOL   ARITHMETIC 

5.  What  is  the  length  of  one  degree  of  a  circumference 
which  measures  720  miles  ? 

6.  From  April  21  to  June  15  is  how  many  days  ? 

7.  Two  quarts  of  alcohol  will  fill  how  many  4-ounce 
bottles  ? 

8.  A  10-acre  field  contains  how  many  square  rods  ? 

9.  What  is  the  altitude  of  a  parallelogram  whose  area  is 
132  sq.  ft.  and  whose  base  is  12  ft.  ? 

10.    What  is  the  area  of  a  triangle  whose  base  is  4  yd.  and 
whose  altitude  is  6  f t.  ? 

646.    Written 

1.  A  wall  77  ft.  long,  6  ft.  high,  and  12  in.  thick  is  built  of 
bricks  costing  f  9  per  M.  What  was  the  entire  cost  of  the  bricks 
if  22  bricks  were  sufficient  to  make  a  cubic  foot  of  wall  ? 

2.  The  altitude  of  a  triangle  is  16  ft.  6  in.,  and  the  base 
30  ft.  6  in.     What  is  the  area? 

3.  The  altitude  of  a  triangle  is  60  ft.  and  the  area  3600  sq. 
ft.     What  is  the  base  ? 

Hint.  —  Let  x  =  the  base,  and  make  an  equation. 

4.  Find  the  cost  of  a  carpet  for  a  floor  15  ft.  square,  if  the 
carpet  is  f  yd.  wide  and  costs  1)1.25  a  yard,  making  no  allow- 
ance for  waste. 

5.  Find  the  cost  of  a  steel  ceiling  for  a  room  18  ft.  6  in.  by 
28  ft.  6  in.,  at  the  rate  of  16  cents  per  square  foot. 

6.  How  much  milk  is  contained  in  83  cans,  each  holding  8 
gal.  2  qt.  1  pt.  ? 

7.  How  much  coal  is  there  in  9  loads  of  2  T.  250  lb.  each  ? 

8.  Find  the  value  of  a  pile  of  4-foot  wood,  40  ft.  long  and 
5  ft.  high,  at  $  5.50  per  cord. 


REVIEW  AND  PRACTICE  379 

9.    Find  the  total  weight  of  three  loads  of  hay  containing  1  T. 
2  cwt.  78  lb.,  1  T.  3  cwt.  39  lb.,  and  19  cwt.  89  lb.,  respectively. 
10.    A  5-gallon  oil  can  lacks  3  qt.  1  pt.  of  being  full.     What 
is  the  value  of  the  oil  in  the  can   at  12^  per  gallon  ? 

647.  Oral 

1.  An  inch  board  containing  6  ft.  of  lumber  is  6  in.  wide. 
How  long  is  it  ? 

2.  A  block  of  wood  1  ft.  square  and  9  in.  thick  contains 
how  many  board  feet  ? 

3.  Draw  a  full-size  picture  of  a  board  foot. 

4.  A  box  5  in.  by  4  in.  by  9  in.  contains  how  many  cubic 
inches  ? 

5.  A  rectangular  tin  can  4  in.  square  has  a  volume  of  96 
cu.  in.      What  is  its  other  dimension  ? 

6.  If  one  man  can  mine  6  tons  of  coal  in  a  10-hour  day,  how 
many  tons  can  he  mine  in  an  8-hour  day,  at  the  same  rate? 

7.  In  what  denominations  is  volume  expressed? 

8.  In  what  denominations  is  capacity  expressed  ? 

9.  Knowing  the  number  of  cubic  inches  in  a  gallon,  how 
may  we  find  the  number  of  cubic  inches  in  a  liquid  quart  ? 

10.    Knowing  the  number  of  cubic  inches  in  a  bushel,  how  may 
we  find  the  number  of  cubic  inches  in  a  dry  quart  ? 

648.  Written 

1.  A.  garden  plot  30  ft.  long  contains  450  sq.  ft.  of  land. 
What  is  the  cost  of  inclosing  it  with  wire  fence  at  27  cents  a 
yard  ? 

2.  Find,  to  the  nearest  tenth,  the  number  of  bushels  of  grain 
that  can  be  stored  in  a  bin  6  ft.  long,  3J  ft.  wide,  and  5  ft.  high. 

3.  What  is  the  weight  of  a  load  of  wheat  that  exactly  fills  a 


380  GRAMMAR  SCHOOL   ARITHMETIC 

wagon  box  tliat  is  14  ft.  long,  3  ft.  wide,  and  20  in.  deep,  the 
weight  of  a  bushel  of  wheat  being  60  lb.  ?  (Answer  correct  to 
tenths'  place.) 

4.  A  rectangular  cistern  is  22  ft.  long  and  7  ft.  wide.  When 
it  contains  32  barrels  of  water,  how  deep  is  the  water? 

5.  Make  out  a  bill  of  four  items  for  goods  bought  at  a  dry- 
goods  store.     Foot  and  receipt  the  bill. 

6.  Make  out  a  statement  of  an  account  at  a  hardware  store, 
using  four  debit  items  and  two  credit  items. 

7.  A  farmer  sold  a  load  of  hay  weighing  1850  lb.  at  i  15  a 
ton,  and  with  a  part  of  the  money  received  bought  1  T.  5  cwt. 
of  coal  at  $  6.20  per  ton.     How  much  money  had  he  left  ? 

8.  Find  the  exact  number  of  days  from  Dec.  9, 1907,  to  June 
30,  1908. 

9.  A  wheel  91  ft.  in  circumference  will  make  how  many 
revolutions  in  going  11  mi.? 

10.    Reduce  -^^  to  a  decimal. 

649.     Oral 

1.  What  rate  per  cent  is  equal  to  ^^^  ;  ^;  ^;  | ;  -|? 

2.  Find  20  %  of  500  lb. ;  331  %  of  60  bu. ;  16|  %  of  f  18. 

3.  What  decimal  is  equivalent  to  |  of  1  %  ? 

4.  What  per  cent  is  equivalent  to  .25?  to  .025?  to  .0025? 

5.  A  gain  of  $  10  on  goods  costing  $  20  is  what  per  cent  gain  ? 

6.  A  gain  of  f  10  on  goods  sold  for  $  20  is  what  per  cent  gain  ? 

7.  What  is  the  selling  price  of  goods  that  cost  $200  and  were 
sold  at  10%  advance? 

8.  What  is  the  cost  of  goods  that  bring  150  when  sold  at  a 
gain  of  25  %  ? 


REVIEW   AND   PRACTICE  381 

9.  What  is  an  agent's  commission  on  ten  books  which  he  sells 
for  $4  apiece  and  receives  40%  commission? 

10.  When  an  agent  sells  goods  at  a  commission  of  20%, 
what  does  his  principal  receive  for  goods  that  the  agent  sells 
for  1200? 

650.    Written 
1.   Add: 
23.75  2.    What   was    the    cost    of    goods    that    brought 

8.679     11120.20  when  sold  at  20  %  profit? 

3    Potatoes  sold  at  10^  per  half  peck  yield  a  profit 
835.406     q£  ^^  cj       ],^ind  the  cost  per  bushel. 
42.973  . 

9  009         *•    ^^^^^  ^^  ^^®  P^^  c®^^  o^  l^^s  o^  ^  house  bought 

80896  ^^^  ^ ^^^^  ^^^^  ^^^^  ^^^  ^ ^^^^ ^ 
7.234         5.    A  merchant  paid  8  900  for  200  bbl.  of  flour.     The 

3.876  freight  cost  him  45/  a  barrel  and  the  cartage  5/  a 

98.423  barrel.      At  what  price  per  barrel  must  he  sell  the 

1.89  flour  to  gain  21%? 

.yo/  g     What  is  the  cost  of  snoods  sold  for  $585  at  a  loss 
2.496  ,f2.i%? 
53.875  ^ 
7.    A  commission  merchant  sold  a  consignment  of 

700  doz.  eggs  at  181/  and  one  of  900   doz.  at  21  j/.     What 
was  the  amount  of  his  commission  at  4|  %  ? 

8.  An  agent  remitted  to  his  principal  12695.10  as  the  net 
proceeds  of  the  sale  of  a  consignment  of  goods,  having  retained 
his  commission  of  5  %,  and  $12.40  for  expenses  incurred.  What 
was  the  amount  of  his  sales? 

9.  The  Kansas  City  agent  of  a  Philadelphia  manufacturer 
receives  an  annual  salary  of  f  2000  and  a  commission  of  2  %  on 
all  his  sales.     His  sales  for  the  month  of  January  amounted  to 


882  GRAMMAR   SCHOOL  ARITHMETIC 

$7329.     If  he  did  as  well  for  the  remainder  of  the  year,  what 
was  his  total  income? 

10.  A  wagon  listed  at  $200  was  bought  by  a  dealer  at  20  and 
10  cjo  off?  3.nd  sold  by  him  at  5  and  10  %  off  from  the  same  list 
price. 

a.    How  much  did  he  gain? 
h.    What  per  cent  did  he  gain? 

651.     Orol 

1.  My  furniture  has  been  insured  12  years  at  the  rate  oi\fjo 
premium  on  a  three-year  policy.  How  much  have  I  paid  on  a 
$1000  policy? 

2.  What  agreement  does  a  man  make  when  he  indorses  a 
note  in  blank  ? 

3.  What  is  the  bank  discount  on  a  60-day  note  for  $100 
without  interest,  if  discounted  at  date  at  the  rate  of  6  %  per 
year?     If  discounted  30  da.  after  date? 

4.  What  would  I  receive  for  my  note  for  $  100  for  90  da., 
without  interest,  if  I  sold  it  to  the  bank  on  the  day  of  date,  the 
discount  rate  being  6%  per  year? 

5.  Why  do  banks  protest  notes  when  they  become  due? 

6.  When  the  tax  rate  is  12  mills  on  the  dollar,  what  is  my 
tax  on  property  assessed  at  $1000? 

7.  What  is  the  meaning  of  "Exchange  on  London,  4.86^"? 
"  Exchange  on  Paris,  5.171"  ?     »  Exchange  on  Hamburg,  97| "  ? 

8.  What  American  coin  is  most  nearly  like  the  mark  ?  the 
franc?  the  sovereign? 

9.  When  the  tax  rate  is  .01,  what  is  the  assessed  valuation 
of  property  on  which  the  tax  is  $120? 

10.  When  the  exchange  value  of  1  mark  is  24^,  what  is  the 
quoted  rate  of  exchange  on  Germany? 


REVIEW  AND   PRACTICE 


383 


652.    Written 

1.  The  report  of  a  savings  bank 
shows  the  following  resources.  Find 
the  total. 

Bonds  and  mortgages      $5,979,120.95 

Bonds  of  states  388,312.50 

Boston  city  bonds  372,937.50 

N"ew  York  City  bonds  956,059.45 

Buffalo  city  bonds  39,800.00 

Syracuse  city  bonds  1,178,637.50 

Bonds  of  other  cities  200,092.50 

Onondaga  county  bonds  65,975.00 

New  York  county  bonds  106,150.00 

Bonds  of  towns  192,514.25 

School  district  bonds  12,315.50 

Railroad  bonds  2,924,466.83 

Banking  house  200,000.00 

Other  real  estate  161,777.91 

Cash  in  banks  312,919.22 

Cash  on  hand  88,421.35 

Interest  accrued  229,800.62 


2.  When  the  county  tax 
rate  is  .004376,  what  is  the 
county  tax  on  property  as- 
sessed at  15000? 

3.  John  Brown  owes  Fred 
Haskins  1200.  Haskins  draws 
on  Brown  for  that  amount, 
making  the  draft  payable  at 
sight  to  the  First  National 
Bank.     Write  the  draft. 

4.  A  factory  worth  149,677 
is  insured  for  |  of  its  value,  at 


1 1  % .     What  is  the  premium? 

5.  1420  premium  on  a  fire 
insurance  policy  of  $  56,000  is 
what  rate  ? 

6.  A  city  whose  population  is  22,000  has  an  assessed  valua- 
tion of  $  11,000,000.  Mr.  Carpenter  owns  a  house  in  that  city 
valued  at  f  2800.  What  was  his  share  of  the  tax  for  building 
a  new  high  school  costing  $  75,000  ? 

7.  Find  the  amount  of  $  867.35  for  1  yr.  8  mo.  27  da.  at 


9%. 

8. 

6  mo. 


What  principal  at  6%  will  amount  to  f  272.50  in  1  yr. 


9.    How  long  will  it  take  I  360  to  gain  $  53.64  at  6  %  ? 
10.    A  man  bought  a  bill  of  lumber  for  I  850,  Jan.  1,  1907, 
giving  his  note  with  interest  at  6%.     He  paid  flOO  May  1, 
and  $  150  Aug.  16.     What  was  due  at  settlement,  Nov.  1,  1907, 
by  the  United  States  rule  ? 


384  GRAMMAR  SCHOOL  ARITHMETIC 

653.    Oral 

1.  a.  Draw  a  line  one  meter  long  without  a  measure. 
Measure  and  correct  it. 

h.    Draw  a  line  80%  of  a  meter  long. 

c.  Draw  a  line  20  %  as  long  as  the  one  in  h. 

d.  The  line  in  c  is  what  per  cent  as  long  as  the  line  in  a  ? 

2.  a.  Without  using  a  measure,  draw  a  square  meter.  A 
square  decimeter. 

h.  Draw  a  line  dividing  the  square  meter  into  two  parts, 
one  of  which  is  four  times  as  large  as  the  other. 

c.  How  many  square  decimeters  are  there  in  each  of  these 
parts  ? 

3.  Estimate  the  number  of  square  meters  in  the  floor  of 
your  class  room. 

4.  Name  some  object  whose  volume  is  about  one  cubic  deci- 
meter.    Its  size  is  like  that  of  what  unit  of  capacity  measure  ? 

5.  a.    One  kilogram  is  about  how  many  pounds  ? 

h.  A  man  bought  a  load  of  coal  weighing  1000  Kg.  About 
how  many  pounds  did  it  weigh  ? 

6.  What  is  the  duty  on  flOO  worth  of  mahogany  boards 
at  15%? 

7.  A  box  5  dm.  long,  3  dm.  wide,  and  2  dm.  deep  will  hold 
how  many  liters  of  oats  ? 

8.  A  cubic  decimeter  of  water  weighs  how  many  grams  ? 

9.  What  is  the  value  of  100  shares  of  bank  stock  quoted 
at  1031? 

10.    How   many  dollars  of  city  bonds    can   be   bought   for 
$104,000,  when  they  are  selling  at  4  %  premium? 


REVIEW   AND  PRACTICE  385 

654.     Written 

1.    Add: 
4763  2.    Find,  in  hectoliters,  the  capacity  of  a  bin  which 

8257      is  9  m.  long,  1  m.  wide,  and  175  cm.  high. 
6039  3.    How  many  kilograms  of  water  will  fill  a  rec- 

^^'^^      tangular  vat  which  is   5  m.  long,  4  m.    wide,  and 
1397      .50  cm.  deep? 

685 
Q107  **    ^^^^  ^^  *^®  duty,  at  35%,  on  a  shipment  of 

_^  fur  coats  invoiced  at  2150  marks,  less  a  trade  dis- 

^,t^      count  of  4  %  ?     (1  mark  =  |.238.) 
1476  < 

gggg  5.    Find,  by  means  of  equations,  three  numbers,  of 

^^•j      which  the  first  is  smaller  by  106  than  the  second,  the 

gg      third  larger  by  22  than  the  second,  and  the  sum  of 

9432      ^^^  three  is  495. 

7943  6.    A  man   in   St.   Paul  wishes   to  send  |386  to 

8(388      liis    family  in    Berlin.      What   is   the    face    of    the 

draft  which  he  can   buy  with  that  sum,  exchange 

being  at  96|  ? 

7.  A  merchant  in  Galveston  owes  a  bill  of  .£47  10s.  in 
Glasgow.  What  must  he  pay  for  a  draft  for  that  amount 
when  exchange  is  at  4.872? 

8.  On  Jan.  1,  1908,  the  stock  of  the  Wampanoag  Mills  was 
quoted  at  92-|-.  What  must  be  invested  in  this  stock,  includ- 
ing brokerage  at  ^  %,  to  secure  238  shares? 

9.  The  Central  Coal  and  Coke  Company  paid  a  dividend  of 
1|%  on  its  common  stock,  Jan.  15,  1908. 

a.  What  is  the  dividend  on  200  shares  ? 

b.  How  many  shares  must  I  own  in  order  to  receive  a  divi- 
dend of  1900? 

10.    What  must  I  invest  in  4|  %  city  bonds  at  par  to  obtain 
an  annual  interest  of  ^675  ? 


386  GRAMMAR  SCHOOL  ARITHMETIC 

655.  Oral 

1.  Draw  a  vertical  line  on  the  blackboard,  cutting  off  33^% 
of  the  board.  Draw  another  line,  cutting  off  25%  of  what 
remains.     What  fraction  of  the  entire  board  is  cut  off  ? 

2.  When  the  dividend  on  5  shares  of  railroad  stock  is  1 25, 
what  is  the  rate  of  dividend  ? 

3.  What  is  the  annual  interest  on  ten  500-dollar  4  %  bonds? 

4.  What  is  the  ratio  of  75  to  3  ? 

5.  What  is  the  number  whose  ratio  to  45  is  J  ? 

6.  7  :  ?  =  J^  ;   ?  :  18  =  3  ;   ?  :  ?  =  6. 

7.  2:4  =  7;?     3:8=1:?     3  :  ?  =  ?  :  12. 

8.  Divide  1 25  among  three  boys  in  the  ratio  of  1,  2,  and  2. 

9.  Divide  77  into  two  parts  having  the  ratio  of  5  to  6. 

10.    Three  numbers  are  in  the  ratio  of  1,  2,  and  3.     The  first 
number  is  7.     Find  the  others. 

656.  Written 
1.    Add 

$385.24         2.    Solve  by  proportion  :     What  is  the  cost  of  a 

17.89     200-acre  farm  at  the  rate  of  25  acres  for  11324  ? 

3.20         3,    What  sum  of  money  will  yield  as  much  interest 

^^^-  in  4  yr.  6  mo.  as  19000  will  yield  in  9  mo.  ? 

831  19  ^                  J 

9m  ^1  *•    -^^^  long  will  it  take  435  men  to  earn  as  much 

„-*   „  money  as  145  men  can  earn  in  4  yr.  3  mo.  ? 

n  gQ  5.    When  a  post  4  ft.  6  in.  high  casts  a  shadow 

98  36  ^  ^^'  ^i  ^^*  ■^^^^'  ^^^  high  is  a  tree  that  casts  a 

521  83  shadow  40  ft.  6  in.  long  ? 

829.17  6.    How  many  Kl.  of  water  can  be  kept  in  a  vat 

743.65  that  is  2^  m.  by  15  dm.  by  50  cm.  ? 

812.79  7.    Two  boys,  having  received  40  cents  for  some 


REVIEW  AND  PRACTICE  387 

work,  divided  it  so  that  one  boy  received  |  as  much  as  the 
other.     How  much  did  each  receive  ? 

8.  C  failed  in  business,  owing  A  13000,  B  12500,  and 
D  14500.  His  property  was  worth  only  $6400.  How  much 
should  each  creditor  receive  ? 

9.  A  farmer  bought  two  cows  for  |80,  paying  |  as  much 
for  one  as  for  the  other.     Find  the  cost  of  each. 

10.    Separate  2723  into  three  parts  having  the  ratio  of  ^  to 
1  to  2. 

657.  Oral 

1.  Find  the  value  of  2*  ;  5^;  3^  ;  7^ ;  5^  ;  2^  ;  122. 

2.  A  number  which  is  the  product  of  equal  factors  is  called 
what? 

3.  Find  the  value  of  VT6  ;   -^16  ;   ^216  ;   V400  ;    \/32. 

4.  Finding  one  of  the  equal  factors  which  produce  a  num- 
ber is  called  what  ? 

5.  The  legs  of  a  right  triangle  are  3  ft.  and  4  ft.  What  is 
the  hypotenuse  ? 

6.  The  hypotenuse  of  a  right  triangle  is  10  ft.,  and  one  of 
the  legs  8  ft.     What  is  the  other  leg  ? 

7.  What  are  the  two  equal  factors  of  121  ? 

8.  Find  one  of  the  three  equal  factors  of  ^. 

9.  One  of  the  three  equal  factors  of  a  number  is  5.  What 
is  the  number  ? 

10.   The  entire  surface  of  a  cube  is  24  sq.  in.     How  long  is 
each  edge  of  the  cube  ? 

658.  Written 

1.    Find  the  square  root  of  3,396,649. 


388  GRAMMAR  SCHOOL   ARITHMETIC 

2.    The  entire  surface  of  a  cube  is  1350  sq.  in.     Find  the 
volume  of  the  cube. 


o        V2IO25.  —    ? 

4.  The  perimeter  of  a  square  is  1320  rd.  Find  its  area 
in  acres. 

5.  How  many  feet  of  fence  are  required  to  inclose  a  square 
field  containing  2|^  A.  ? 

6.  A  cylindrical  oil  tank,  24  ft.  in  diameter  and  18  ft.  high, 
will  contain  how  many  barrels  of  oil,  allowing  4^  cu.  ft.  for  a 
barrel  ? 

7.  Find,  to  the  nearest  tenth  of  a  foot,  the  depth  of  a  cylin- 
drical cistern  whose  capacity  is  40  barrels,  and  the  diameter  of 
whose  base  is  6  ft. 

8.  a.  Find  the  difference  in  time  between  two  places  in  79° 
18' '  and  103°  4-'  west  longitude,  respectively. 

h.  When  it  is  noon  at  the  first  place,  what  is  the  time  at  the 
second  place  ? 

9.  When  it  is  11  a.m.  at  a  place  in  73°  1"  west  longitude, 
what  is  the  time  at  a  place  in  14°  53^'  east  longitude  ? 

10.  What  is  the  longitude  of  a  place  in  which  the  time  is 
half -past  one  a.m.,  when  it  is  midnight  at  a  place  whose  longi- 
tude is  47°  17'  15''  East  ? 

659.  1.  A  coal  company  has  $85,000  invested  in  a  shaft 
mine.  Assuming  the  cost  of  mining  the  coal  and  preparing  it 
for  market  to  be  76^  per  ton,  the  average  price  received  to  be 
$1.05,  and  the  commission  paid  for  selling  to  be  5^  per  ton, 
how  many  tons  per  year  must  the  company  take  from  this  mine 
to  yield  a  net  income  of  8  %  on  the  investment  ? 

2.  A  mine  owner  bought  coal  at  $2130  per  acre  and  mined 
it.     The  vein  averaged  5  ft.  6  in.  in   thickness   and  yielded 


REVIEW  AND  PRACTICE  389 

1000  tons  of  coal  per  acre  for  each  foot  of  the  thickness  of  the 
vein.  If  the  net  price  received  for  the  coal  was  98^  per  ton, 
what  was  received  for  7|  acres  of  the  coal  ? 

3.  A  pane  of  plate  glass  was  listed  at  $96.40,  with  trade 
discounts  of  75  and  5%,  and  a  further  discount  of  2%  for  cash 
payment.     What  was  the  net  cash  price  ? 

4.  Make  out  and  receipt  a  bill  for  22J  yd.  of  muslin  at  14^ 
per  yard,  5|  yd.  of  cambric  at  12^  a  yard,  and  20  handkerchiefs 
at  f  3.60  per  dozen. 

5.  A  typist  writes  daily  130  folios  of  10  lines  each,  averag- 
ing 10  words  to  a  line  and  7  letters  to  a  word.  Her  typewriter 
has  42  keys,  5  of  which  are  vowel  keys.  If  the  vowel  keys  are 
used  three  times  as  often  as  the  other  keys,  how  many  vowels 
are  written  in  a  day  ? 

6.  When  camphor  gum  is  bought  at  85^  per  pound  and  sold 
at  10/  an  ounce  Avoirdupois,  what  is  the  rate  per  cent  of  profit  ? 

7.  A  druggist  who  buys  cocaine  at  the  rate  of  $5  per  ounce 
of  480  gr.  and  sells  it  at  the  rate  of  2  gr.  for  5/,  gains  what 
per  cent  ? 

8.  The  railroad  company  charges  §59.40  for  the  use  of  a 
freight  car  from  Quincy,  Mass.,  to  Syracuse,  N.  Y.,  and  is  re- 
sponsible for  all  damages  to  the  freight  carried.  Mr.  Harding, 
by  releasing  the  company  from  liability  for  damage,  secured  a 
reduction  of  331%  from  the  regular  freight  rate.  He  then  had 
his  freight  insured  for  $2000,  at  a  premium  rate  of  -1%.  How 
much  did  he  save  on  a  carload  of  freight  by  this  plan  ? 

9.  Simplify  ^      ^~\    ^  and  express  the  result  as  a  decimal. 

10.  Find  the  sum  which  a  bank  would  pay  for  a  note  for 
1750,  without  interest,  90  da.  before  it  was  due,  if  its  discount 
rate  was  7  %  per  annum. 


390  GRAMMAR  SCHOOL   ARITHMETIC 

660.    1.    Add: 

$289.52  Note.  —  Problems  2-6  are  taken  from  an  arithmetic  published 

rjq  nn      over  one  hundred  years  ago. 

81.73  2.    There  are  two  numbers  ;    the   less  number  is 

786.39  8761,  the  difference  between  the   numbers   is   597. 

496.38  What  is  the  sum  of  the  numbers? 

809.99  3.    What  is  the  length  of  the  road,  which,  being 

78.63  33  ft.  wide,  contains  an  acre? 

61.92  ^     ^  bankrupt  whose  effects  are  $3948  can  pay 

^•^^  his  creditors  but  28  cents  5  mills  on  the  dollar.    What 

689.73  does  he  owe? 

^QQ  no 

5.    The  river  Po  is  1000  feet  broad  and  10  feet 

deep,  and  it  runs  at  the  rate  of  4  miles  an  hour.     In 

QQ*8^     what  time  will  it  discharge  a  cubic  mile   of  water 

«R*zlQ      (reckoning  5000  feet  to  the  mile)  into  the  sea  ? 

808.70         ^'    ^^  ^^®  ^^^^  census,  taken  a.d.  1800,  the  num- 
ber  of  inhabitants  in  the  New  England  states  was  as 
follows,  viz.:  New  Hampshire,  183,858;  Massachu- 
setts, 422,845  ;  Maine,   151,719  ;   Rhode  Island,   69,122  ;    Con- 
necticut, 151,002  ;  Vermont,  154,465.     What  was  the   entire 
number  ? 

7.  Draw  two  straight  lines  having  the  ratio  of  3  to  2. 

8.  What  is  the  selling  price  of  48  yd.  of  cloth  bought  at 


604.52 
900.68 


I.  per  yard  and  sold  at  a  gain  of  21| 


9.  Estimating  a  bushel  of  coal  to  weigh  80  lb.,  find  to  the 
nearest  tenth  the  number  of  cubic  feet  of  space  needed  for  the 
storage  of  one  ton  of  coal. 

10.    Find  the  product  of  the  common  prime  factors  of  1395 
and  1736. 

661.    1.    4937x398=? 


KEVIEW  AND  PRACTICE  391 

2.  A  note  drawn  for  90  da.  without  interest  was  discounted 
24  da.  after  date,  at  6%  per  annum,  yielding  $553.84  proceeds. 
What  was  the  face  of  the  note  ? 

3.  a.  How  many  kiloliters  of  water  can  be  contained  in  a 
rectangular  cistern  2.5  m.  by  3.6  m.  and  75  cm.  deep? 

h.    What  is  the  weight  of  this  water  in  kilograms  ? 

4.  a.  How  many  shares  of  preferred  stock,  paying  b\%  divi- 
dends, must  I  buy  to  secure  an  annual  income  of  $500.50  ? 

h.    What  will  the  stock  cost,  at  124|,  brokerage  -1%  ? 

5.  A  barn  roof  is  58  ft.  long  and  the  slant  height  is  24  ft. 
on  each  side.  Find  the  cost  of  the  shingles  for  this  roof  at 
$5.00  per  M,  allowing  1000  shingles  for  120  square  feet. 

6.  When  it  is  noon  at  Boston,  71°  4'  west  longitude,  what 
is  the  time  at  Rochester,  77°  51'  west  longitude  ? 

7.  a.  A  six  months'  note  for  $900  without  interest,  dated 
Oct.  26,  1906,  is  discounted  Feb.  21,  1907,  at  6%.  What  are 
the  proceeds  ? 

h.    If  the   note  were  interest-bearing,  what  would  be   the 
proceeds  ? 

8.  A  tract  of  land  is  424  rods  long  and  324  rods  wide.  It 
cost  $36919.80.     What  was  the  cost  per  acre  ? 

9.  Three  loads  of  coal  weighing  respectively  3805  lb.,  3965 
lb.,  and  4730  lb.,  cost  $38.75.     What  was  the  price  per  ton  ? 

10.    Find  the  square  root  of  160  correct  to  four  decimal 
places. 


APPENDIX 

CUBE  ROOT 

The  cube  of  a  number  composed  of  tens  and  units  may  be  found 
as  follows : 

24  =  20  +  4  =  2  tens  +  4  units. 

243  =  (20  +  4)  X  (20  +  4)  X  (20  +  4). 

20  +  4=       24 
20  +  4  =        24 


(20  X  4)  +  4-^  =       96 
202  +  (20  X  4)  =      480 


202  +  2  X  (20  X  4)  +  42  =      576 
20  +  4  =        24 


(202  X  4)  +  2  X  (20  X  42)  +  43  =    2304 
203  +  2  X  (202  X  4)  +  (20  X  42)  =  11520 

203  _^  3  X  (202  X  4)  +  3  X  (20  X  42)  +  43  ="l3824 

From  the  operation  we  find  that, 

The  cube  of  the  tens 203  =  8000 

3  times  the  square  of  tens  multiplied  by  units       .     .     .  3  x  (202  x  4)  =  4800 

3  times  the  tens  multiplied  by  the  square  of  the  units    .  3  x  (20  x  42)  =z  960 

The  cube  of  the  units 43  =  64 


8000  +  4800  +  960  +  64  =  13824 
Summary 

The  cube  of  a  number  composed  of  tens  and  units  is  equal  to  the  cube 
of  the  tens  plus  3  times  the  square  of  the  tens  multiplied  by  the  units, 
plus  3  times  the  tens  multiplied  by  the  square  of  the  units,  plus  the  cube 
of  the  units. 

By  reversing  the  process,  we  may  find  the  cube  root. 

1.    What  is  the  cube  root  of  13,824  ? 

Solution.  —  Separating  into  periods  of  three  figures  each,  beginning  at  units, 
we  have  13'824.  Since  there  are  two  periods  in  the  power,  there  must  be  two 
figures  in  the  root,  tens  and  units. 

392 


APPENDIX  393 

The  greatest  cube  of  tens  contained  in  13824  is  8000,  and  its  cube  root  is  20 
(2  tens). 

13'824  I  20  +  4 
Tens3  =  203=  8000 

3  X  tens2  =  3  x  202  =  1200     6824 
3  X  tens  x  units  =  3  x  20  x  4  =    240 
units'^  =  42  =      16 
3  X  tens2  +  3  tens  x  units  +  units2  =  1466 
(3  X  tens2  4-  3  X  tens  x  units  +  units2)  x  units  =  6824 

Subtracting  the  cube  of  the  tens,  8000,  the  remainder,  6824,  consists  of  3  x 
(tens2  X  units)  +  3  x  (tens  x  units2)  +  units^.  6824,  therefore,  is  composed  of 
two  factors,  units  being  one  of  them,  and  3  x  tens2  +  3  x  tens  x  units  +  units2, 
being  the  other.  But  the  greater  part  of  this  factor  is  3  x  tens2.  By  trial  we 
divide  5824  by  3  x  tens^  (1200)  to  find  the  other  factor  (units),  which  is  4  if 
correct.  Completing  the  divisor,  we  have  12002  _j_  3  x  (20  +  4)  +  42  =  1456, 
which,  multiplied  by  the  units,  4,  gives  the  product,  6824,  proving  the  correct- 
ness of  the  work.     Therefore,  the  cube  root  is  20  +  4,  or  24. 

To  find  the  cube  root  by  the  aid  of  blocks. 

Finding  the  cube  root  of  a  number  is  equivalent  to  finding  the 
thickness  of  a  cube,  its  volume  being  given. 

The  following  formulas  illustrate  the  principles  that  underlie 
operations  in  cube  root. 

Note.  — For  convenience,  I,  6,  t,  and  v  will  represent  length,  breadth,  thick- 
ness, and  volume,  respectively. 

(1)    lxbxt  =  v.  (2)   v-^{lxh)  =  U  (3)   v-^(lxt)=h. 

(4)   v^(hxt)  =  l. 

2.  What  is  the  thickness  of  a  cube  whose  volume  is  13824  cubic 
feet? 


Solution.  —  The  greatest  cube  of 
even  tens  contained  in  13824  cu.  ft.  is 
8000  cu.  ft.  (Cube^,p.394.)  Its  thick- 
ness, therefore,  is  20  ft.  Subtracting 
8000  {A)  from  13824  leaves  a  re- 
mainder of  5824  cu.  ft. ,  which  are  added  in  solids  of  equal  thickness  to  three  sides  of 
A,  as  seen  in  Fig.  2.    It  now  remains  to  find  the  thickness  of  the  additions  (6,  c,  d) , 


AND  BREADTH 

VOLUME 

THICKNESS 

3  x  202           ^  1200 

13'824 

20  ft. 

2  X  20   X  4  =    240 

8000 

4  ft. 

42=      16 

6824 

24  ft. 

1466 

.      6824 

394 


GRAMMAR  SCHOOL  ARITHMETIC 


(e, /,  g)^  and  ^,  which  have  a  uniform  thickness.  As  the  solids,  6,  c,  d,  form 
the  greater  part  of  the  volume  of  the  additions  (5824  cu.  ft.),  and  the  length  and 
breadth  of  each  is  20  ft.  (the  length  and  breadth  of  yl),  by  trial,  using  Formula 
2,  we  find  5824  ^  (3  x  20'^)  =  4  f t. ,  thickness  of  the  additions,  if  correct.  Know- 
ing the  thickness,  which  is  also  the  breadth  of  e,/,  g,  h,  we  find  the  product  of  the 
length  and  breadth  of  e,  /,  ^  =  3  x  20  x  4  =  240  sq!  ft. ;  and  that  of  7i  =  42  =  16 
sq.  ft.  ;  both  of  which  added  to  1200  sq.  ft.  =  the  product  of  the  length  and 


■^ 


Fig.  2. 


^ 


^ 


IS 

h 


^ 


^ 


"^ 


^ 


9 


breadth  of  all  the  additions.    This  product,  by  Formula  1,  multiplied  by  the 
thickness,  4  ft.  =  5824  cu.  ft.,  proving  the  correctness.     Therefore, 

The  thickness  of  a  cube  whose  volume  is  13824  cu.  ft,  is  20  -f  4  ft.,  or  24  ft. 
The  numbers  in  the  middle  column  (Ex.  2)  all  indicate  volume  : 
13824  =  volume  of  original  cube. 
8000  =  volume  of  Cube  A. 

5824  =  volume  of  the  additions  (6,  c,  d),  (e,  /,  g),  and  h. 
The  numbers  in  the  left-hand  column  indicate  product  of  length  and  breadth: 
1200  =1  xboi  solids  &,  c,  d. 
240  =  Z  X  6  of  solids  e,  /,  g. 
16  =  Z  X  6  of  cube  h. 

The  numbers  in  the  right-hand  column  indicate  thickness : 
20  ft.  =  thickness  of  A. 
4  ft.  =  thickness  of  all  the  additions. 
24  ft.  =  thickness  of  original  cube. 


APPENDIX 


395 


Short  method. 
Rule  for  finding  the  cube  root: 

Beginning  at  the  decimal  point,  separate  the  number  into  periods  of 

three  figures  each,  thus:  16 '581'.  375. 
Find  the  greatest  cube  in  the  left-hand  period,  and  write  its  root  at 

the  right.     Subtract  the  cube  from  the  left-hand  period,  and  bring 

down  the  next  period  for  a  dividend,  thus : 

16'581'.375|2 

8 

8581 

To  find  the  trial  divisor*,  square  the  root  already  found  with  a  cipher 
annexed,  and  multiply  by  3,  thus : 

16^581'.876  |_2  20 


8 


Trial  divisor,  1200)  8581 


20 
400 
3 

1200 


To  find  the  trial  figure,  find  how  many  times  the  trial  divisor  is  con- 
tained in  the  dividend,  thus : 

16'581'.375  [25  20 


8 


Trial  divisor,  1200)  8581 


20 
400 

3 

1200 


To  find  the  correction,  multiply  the  former  root  by  S,  annex  the  trial 
figure,  and  multiply  by  the  trial  figure,  thus : 

2 

66 
_5 
825 

Continue  thus,  until 
all  the  periods  are  ex- 
hausted. 

Note  1.  —  When  there  is  a  remainder  after  all  the  periods  are  exhausted,  an- 
nex decimal  periods,  and  continue  the  process  as  far  as  desired.  The  result  will 
be  the  approximate  root. 


1200 

325 

1525 

16'581'.375 
8 

|25.6 

divisor. 

8581 
7625 

187500 

3775 

191275 

956376 
956375 

396 


GRAMMAR  SCHOOL   ARITHMETIC 


Note  2.  —  When  a  cipher  occurs  in  the  root,  we  annex  two  ciphers  to  the  tria^ 
divisor,  and  bring  down  the  next  period. 

Note  3.  —  The  right-hand  decimal  period  must  have  three  places. 

3.   What  is  the  cube  root  of  8.414975304? 


Operation. 


8.414'975'304  I  2.034 


120000 
1809 

414975 

121809 

365427 

6362700 

24376 

12387076 

49548304 
49548304 

Since  0  occurs  in  the  root,  annex 
00  to  the  trial  divisor,  making 
120000 ;  bring  down  the  next  pe- 
riod. 


Note.  —  To  find  the  cube  root  of  a  common  fraction,  extract  the  root  of  each 
term  separately.  If  both  terms  are  not  cubes,  reduce  to  a  decimal  and  then 
extract  the  root.    The  result  will  be  the  approximate  root. 


Find  the  cube  root  of : 

4.  42875 

5.  884736 

6.  4492125 

7.  77854483 

8.  8.615125 


9.  17.373979 

10.  450827 

11.  1879.080904 

12.  32.890033664 

13.  10077696 


14.  What  is  the  cube  root  of  ||f|fi  ?  j\ ?  j^}^?  Sd^\?  ^^? 
Extract  the  cube  root  to  the  third  decimal  place : 

15.  14.323  17.    .06324  19.  3 

16.  31982.4  18.    .0015  20.  7 

21.  What  is  the  width  of  a  cube  whose  volume  is  91125  cubic 
inches  ? 

22.  A  cubical  cistern  holds  50  barrels  of  water.     How  deep  is  it? 

23.  What  is  the  entire  surface  of  a  cube  whose  edge  is  9  ft.? 

24.  a/.006  x32.5  =  ? 


APPENDIX  397 

SIMILAR  SOLIDS 

Solids  having  the  same  form  without  regard  to  size  are  similar  solids. 
Any  two  cubes  or  any  two  spheres  are  similar  solids.  Solids  are 
similar  when  their  corresponding  dimensions  are  proportional. 

Similar  solids  are  to  each  other  as  the  cubes  of  their  correspond- 
ing dimensions. 

1.  A  globe  is  3  inches  in  diameter,  and  another  6  inches  in  diam- 
eter.    What  is  the  ratio  of  their  volumes  ? 

Explanation.  —  They  are  to  each  other  as  S^  to  6^,  or  27  :  216. 

2.  There  are  64  cubic  inches  in  a  4-inch  cube.  How  many  in  an 
8-inch  cube  ? 

3.  Two  similar  solids  contain  386  and  284  cubic  inches,  respec- 
tively.    If  the  larger  is  11  inches  thick,  how  thick  is  the  smaller  ? 

4.  If  a  man  6  ft.  2  in.  tall  weighs  215  pounds,  what  should  be  the 
weight  of  a  man  5  ft.  10  in.  tall  of  the  same  proportions  ? 

5.  The  width  of  a  bin  is  4  ft.  6  in.  How  wide  must  a  similar  bin 
be  to  hold  4  times  as  much  ? 

6.  An  oil  tank  22  ft.  in  diameter  holds  30,000  gallons. 

a.  How  many  gallons  will  a  tank  of  the  same  shape  and  88  feet  in 
diameter  hold  ? 

h.  What  must  be  the  diameter  of  a  similar  tank  to  hold  3750  gallons  ? 

METHODS   OF  COMPUTING  INTEREST 
METHOD  BY  ALIQUOT  PARTS 
What  is  the  interest  on  f  348  for  3  yr.  5  mo.  15  da.  at  5  %  ? 

$34.80  Interest  for  2  yr.  at  5  %  (yV  of  $348) 
17.40  Interest  for  1  yr.  at  5  %  (i  of  $34.80) 
5.80  Interest  for  4  mo.  at  5  %  (i  of  $  17.40) 
1.45  Interest  for  1  mo.  at  5  %  (}  of  $5.80) 
.73  Interest  for  15  da.  at  5  %  (i  of  $1.45) 
$60.18  Interest  for  3  yr.  5  mo.  15  da.  at  5  %.     Ans. 

If  the  time  were  7  mo.  18  da.,  we  should  separate  it  as  follows  :  (i  of  1  yr.) 
+  (I  of  6  mo.)  -H  (i  of  1  mo.)  +  (^  of  16  da.). 


398  GRAMMAR  SCHOOL  ARITHMETIC 

BANKERS'  METHOD 

This  method  is  variously  known  as  the  Six  per  cent  Bankers^  Sixty 
Day,  Ttvo  Month,  or  Two  Hundred  Month  method.  It  is  based  on 
the  fact  that  any  sum,  on  interest  at  Qt^o,  doubles  in  200  months.  That 
is  to  say,  the  simple  interest  for  200  months  at  Q(fo  i'^  equal  to  the 
principal. 

The  interest  for  2  mo.  is  what  part  of  the  principal  ? 
The  interest  for  6  da.  is  what  part  of  the  principal  ? 

What  is  the  interest  on  $  476  for  2  mo.  19  da.  at  5  %  ? 
$4.76  Interest  for  2  mo.  at  6  %   (yio  of  $476) 
1.19  Interest  for  15  da.  at  6  ^  {\  of  $4.76) 
.24  Interest  for  3  da.  at  6%  (i  of  $1.19) 
.08  Interest  for  1  da.  at  6  %  (i  of  $  .24) 
$  6.27     Interest  for  2  mo.  19  da.  at  6  % 
1.045  Interest  for  2  mo.  19  da.  at  1  % 
$5,225  Interest  for  2  mo.  19  da.  at  5  %     Ans. 

Note. — This  method  is  especially  useful  in  computing  interest  at  6%,  for 
periods  of  90  days  or  less,  a  common  rate  and  time  in  bank  transactions. 

ORDINARY  SIX  PER  CENT  METHOD 

What  is  the  interest  of  $50.24  at  6  %  for  2  yr.  8  mo.  18  da.  ? 

The  interest  of  $1  for  2  yr.     =    2  x  $.06  =  $.12 

for  8  mo.  =    8  x  $.00i  =    .04 

for  18  da.  =  18  x  $.000^  =    .003 

The  interest  of  $  1  for  2  yr.  8  mo.  18  da.  =  $.163 

The  interest  of  $  50.24  is  50.24  times  $.163  =$8.19 

TRUE  DISCOUNT  AND  PRESENT  WORTH 

The  present  worth  of  d  debt  due  at  a  future  time  without  interest  is 
a  sum  which  will  amount  to  the  debt  if  put  at  interest  till  that  time. 

The  debt  is  therefore  the  amount  of  the  present  worth  for  the 
given  time. 

TJie  true  discount  is  the  difference  between  the  debt  and  its  present 
worth.     It  is  the  interest  of  the  present  worth  for  the  given  time. 


APPENDIX  399 

1.  What  is  the  present  worth  and  the  true  discount  of  a  debt  of 
$582.40,  due  in  8  months  without  interest,  when  money  is  worth 

Solution.  —  $  1.04  =  amount  of  $  1  for  8  mo.  at  6  %. 
Statement  op  Relation.  —  $1.04  x  present  worth  =  $582.40. 

$  682.40  -i-  $  1.04  =  $560,  present  worth    )    . 

$  582.40  -  $  560  =  $  22.40,  true  discount )         * 

Summary 

To  find  the  present  worth,  divide  the  face  of  the  debt  by  the  amount 
of$l  for  the  given  time. 

To  find  the  true  discount,  subtract  the  present  worth  from  the  face  of 
the  debt. 

2.  What  are  the  present  worth  and  true  discount  of  $  400,  due 
in  one  year,  when  money  is  worth  5  %  ? 

3.  A  father  wills  his  two  sons  $3000  each,  to  be  paid  in  three 
years  from  the  time  of  his  death.  What  is  the  present  value  of  the 
legacies  if  money  is  worth  6  %  ? 

4.  What  is  the  present  worth  of  f  450,  due  in  two  years  at  5  %  ? 

5.  What  is  the  present  worth  of  $250.51,  payable  in  8  months. 


? 


money  being  worth  6 

6.  Which  is  better,  to  buy  flour  for  $5  cash,  or  for  $5.25  on 
6  months'  time,  when  money  can  be  borrowed  at  5  %  ? 

7.  Find  the  present  worth  of  $  750  for  6  months,  money  being 
worth  6  % . 

8.  What  is  the  present  worth  of  $600,  due  in  1  year  without 
interest,  money  being  worth  6  %  ? 

9.  Write  the  note  which  would  be  given  for  the  above  debt. 

10.  A  man  wishing  to  buy  a  house  and  lot  has  his  choice  between 
paying  $5400  in  cash,  or  $4000  in  cash  and  $1700  in  two  years. 
With  money  at  6  %,  which  is  the  most  advantageous  for  him  ? 

11.  What  is  the  present  worth  of  a  debt  of  $385.31,  due  in  5 
months  15  days,  at  6  %  ? 


400  GRAMMAR  SCHOOL  ARITHMETIC 

12.  Which  would  be  better,  and  how  much,  to  pay  $4000  cash 
for  a  house,  or  $4374.93  in  3  yr.  6  mo.,  money  being  worth  7  %  ? 

13.  I  can  sell  my  house  for  $2800  cash,  or  $3000  and  wait  6 
months  without  interest.  I  choose  the  latter.  Do  I  gain  or  lose, 
and  how  much,  money  being  worth  6  %  ? 

SURETYSHIP 

Suretyship  is  a  contract  whereby  one  party  (usually  a  Jidelity  or 
surety  company)  binds  itself  to  indemnify  another  party  (usually  a  per- 
son or  corporation  employing  some  one  in  a  position  of  trust  or  conji- 
dence)  against  loss  by  the  dishonesty,  willful  neglect,  or  misconduct  of 
an  employee;  or  an  agreement  to  indemnify  one  party  to  a  contract 
against  loss  due  to  the  failure  of  the  other  party  to  fulfill  the  contract. 

The  instrument  by  which  a  contract  of  suretyship  is  made  is  called  a 
suretyship  bond. 

Suretyship  bonds  are  generally  required  of  employees  in  banks ;  collectors 
and  cashiers  ;  treasurers  of  companies,  societies,  cities,  villages,  towns,  coun- 
ties, and  other  political  divisions  of  the  country ;  administrators  of  estates ; 
guardians  of  infants  ;  managers  of  business  enterprises  for  others ;  and  persons 
in  many  other  positions  of  trust.  Contractors  in  all  sorts  of  undertakings  are 
often  required  to  furnish  bonds  for  the  proper  fulfillment  of  their  contracts. 

Suretyship  is  generally  classed  as  a  branch  of  insurance  ;  but  it  differs  from 
ordinary  insurance  in  that  the  surety  receives  its  premium /or  services  rendered, 
rather  than  for  risk  assumed;  and  that  it  does  not  expect  losses  when  executing 
its  b6nd.  Losses  occur  occasionally  from  causes  unforeseen  by  the  surety,  or 
arising  after  the  execution  of  the  bond. 

The  company,  before  issuing  a  suretyship  bond,  inquires  into  the  character 
of  the  person  applying  for  the  bond,  his  habits,  business  standing,  and  reputa- 
tion.    If  these  are  not  satisfactory,  the  application  is  refused. 

A  suretyship  bond  always  involves  three  parties : 

a.  The  principal,  or  party  required  to  furnish  the  bond,  (The  "Em- 
ployee," in  the  bond  given  on  page  401.) 

b.  The  surety,  or  party  joining  ivith  the  principal  in  an  agreement  to 
pay  indemnity  for  loss.     (The  Surety  Company.) 

c.  The  obligee,  or  party  to  whom  the  indemnity  is  promised.  (The 
"Bank"  in  the  bond  given  on  page  401.) 

The  premium  paid  to  the  surety  company  is  computed  at  a  certain 
sum  for  each  one  thousand  dollars  of  the  bond. 


APPENDIX  401 

The  following  form  illustrates  the  essential  parts  of  one  kind  of 
Suretyship   Bond 


Hmerican  Surety  Company 

/imouQt,  $2000  premium,  $8.00 

T^^,  John  Doe,  as  principal,  hereinafter  called  the  "Employee," 
and  the  American  Surety  Company  of  New  York,  hereinafter 
called  the  "Surety  Company,"  as  surety  {in  consideration  of  the  pay- 
ment of  an  agreed  premium  to  it,  the  said  "Surety  Company  "),  bind 
ourselves  for  the  term  commencing  January  1,  1908,  at  9  A.M.,  and 
ending  January  1,  1909,  at  the  like  hour,  to  pay  the  Exchange 
National  Bank  of  St.  Louis,  hereinafter  called  the  "Employer,"  at 
the  home  office  of  the  Surety  Company  in  the  City  of  New  York,  such 
direct  pecuniary  loss  not  exceeding  two  thousand  dollars,  as  it  may 
sustain  of  moneys,  bullion,  funds,  bills  of  exchange,  acceptances, 
notes,  bonds,  drafts,  mortgages,  or  other  valuable  securities  of  similar 
nature,  embezzled,  wrongfully  abstracted  *  *  *  in  the  course  of 
his  employment  as  Messenger  of  said  employer. 

********* 
In  witness  whereof,  said  Employee,  as  principal,  has  hereunto  set 
his  hand  and  seal,  and  mid  American  Surety  Company  of  New  York, 
as  Surety,  has  caused  the  execution  hereof  by  its  President,  and  Assist- 
ant Secretary,  and  its  seal  to  be  hereunto  affixed,  at  the  City  of  New 
York,  this  24th  day  of  December,  1907. 

John  Doe,  Principal  [l.5.] 
American  Surety  Company  of  New  York  [l.5.] 

Richard  Roe,  President 
Attest:  Herbert  Hookway,  Assistant  Secretary 


Oral 

1.    Who  is  the  principal  in  the  above  bond  ?     The  surety  ?     The 
obligee  ? 


402  GRAMMAR   SCHOOL  ARITHMETIC 

•    2.    How  much  is  the  premium  ?     Who  pays  it  ?     Who  receives 
it  ?     What  is  the  rate  per  $1000  ? 

3.  If  John  Doe  remains  in  the  employ  of  the  bank  as  messenger 
for  five  years,  how  much  will  his  bond  cost  him,  during  that  time, 
at  the  same  rate  ? 

4.  How  many  parties  are  there  in  a  suretyship  bond  ?  Name 
them. 

5.  How  many  parties  are  there  in  an  insurance  contract  ?  Name 
them. 

Written 

1-  The  treasurer  of  a  bank  gave  a  bond  of  $  45,000.  What  did 
the  premium  amount  to,  the  rate  being  $  4  per  $  1000  ? 

2.  A  firm  in  Chicago  employed  a  man  to  manage  a  branch  store  in 
Cleveland,  requiring  him  to  give  a  bond  for  $  7500.  He  had  to  pay 
a  premium  of  $56.25  per  year.     What  was  the  rate  per  $1000  ? 

3.  A  cashier  procured  a  bond,  paying  $  5  per  $  1000.  The  pre- 
mium amounted  to  $  22.50.     What  was  the  amount  of  the  bond  ? 

4.  A  tax  collector  held  office  for  four  years,  giving  a  new  bond 
each  year.  He  paid  $  56  in  premiums  during  the  four  years  on  a 
bond  of  $  3500.     What  was  the  rate  per  year  on  $  1000  ? 

5.  A  contractor  furnishing  supplies  to  a  large  manufacturing  con- 
cern gave  a  bond  of  $12,000  for  the  faithful  performance  of  his 
contract.  What  did  he  pay  in  premiums  during  six  years,  the  rate 
being  i  %  per  year  ? 

6.  A  paving  contractor  gave  a  five-year  bond  for  the  fulfillment 
of  his  contract,  paying  a  premium  of  yL  %  per  year.  What  was  the 
auiounii  of  the  bond,  if  the  premium  amounted  to  $  150  ? 

COMPOUND  PROPORTION 

An  equality  between  a  compound  and  a  simple  ratio  is  a  compound 
proportion  J  thus,  * 

'       [■ : :  12 :  20  is  a  compound  proportion. 
o  1 10 ) 


APPENDIX  403 

!Find  the  fourth  terra. 

o.  a)  Solution.  -^  First  change  to  a  simple  proportion,  we  have, 

^'/iy::3:X  3  x  4:6  x  8:  :3:a;. 

^  •  "  Then  divide  the  product  of  the  means  by  the  given  extreme, 

using  cancellation.    Thus, 

2 

1M^  =  12.     Arts. 

1.  If  5  men  earn  $  72  in  8  days,  how  much  can  10  men  earn  in  6 
days? 

Solution.  —  Since  the  answer  is  to  be  in  dollars,  place  $  72  for  the  third  term, 
and  arrange  the  terms  of  each  couplet  according  as  the  answer  should  be  greater 

or  less  than  the  third  term  if  it  depended 
5  men :  10  men  1     .  <«>  79  .  /  \  on  that  couplet  alone. 

8  days  :  6  days  r  •  •  '«'  '  ^  •  W  Since  5  men  earn  $72,  10  men  can 

-'  earn  more,  so  we  place  10  men  for  the 

second  and  5  men  for  the  first ;  and  since  they  earn  $  72  in  8  days,  they  will 
earn  less  in  6  days,  so  we  place  6  days  for  the  second  term,  and  8  days  for  the 
first.     Dividing  the  product  of  the  means  by  the  extremes,  we  have, 

9        2 
$/^x;0x  6  =  $108.    Ans. 

Summary 

Consider  the  answer  as  the  fourth  term,  and  place  the  number  that 
is  like  it  for  the  third. 

Arrange  the  couplets  as  if  the  answer  depended  on  each  couplet  alone, 
as  in  simple  proportion. 

Divide  the  prodiict  of  the  means  by  the  product  of  the  extremes.  Can- 
cel when  possible, 

2.  If  four  horses  eat  10  bushels  of  oats  in  5  days,  hovjr  many 
bushels  will  be  required  to  feed  5  horses  for  2  days  ? 

3.  If  10  men  working  8  hours  a  day  can  do  a  piece  of  work  in  12 
days,  how  many  days  would  it  take  6  men,  working  10  hours  a  day, 
to  do  the  same  amount  of  work  ? 

4.  If  it  costs  $  84  to  carpet  a  room  24  ft.  long  and  21  ft.  wide 
with  carpet  1  yard  wide,  how  much  will  it  cost  to  carpet  a  room  25 
ft.  long  and  12  ft.  wide  with  carpet  27  inches  wide  ? 


404  GRAMMAR  SCHOOL  ARITHMETIC 

5.  If  a  wheelman  rides  144  miles  in  3  days  of  6  hours  each,  how 
many  miles  can  he  ride  in  5  days  of  9  hours  each  ? 

6.  A  section  of  street  33  ft.  long  and  20  ft.  wide  can  be  paved 
with  15,840  stones,  each  9  inches  long  and  8  inches  wide.  How  many 
stones  12  inches  long  and  10  inches  wide  will  it  take  to  pave  a  street 
12  rods  long  and  16  ft.  wide  ? 

7.  If  18  men  chop  360  cords  of  wood  in  12  days  of  9  hours  each, 
how  many  cords  could  17  men  chop  in  13  days  of  10  hours  each  ? 

8.  If  50  men,  working  10  hours  a  day  for  11  days,  can  dig  25  rods 
of  a  canal  60  ft.  wide,  and  5  ft.  deep,  how  many  rods  of  a  canal 
90  ft.  wide,  and  7  ft.  deep,  can  140  men  dig  in  22  days  of  8  hours 
each? 

9.  If  60  men  can  build  a  wall  150  ft.  long,  64  ft.  high,  2  ft.  thick, 
in  8  days  of  10  hours  each,  how  many  days  of  8  hours  each  will  36 
men  require  to  build  a  wall  180  ft.  long,  80  ft.  high,  2i-  ft.  thick  ? 

10.  How  many  men  will  it  require  to  mow  48  acres  in  3  days  of 
12  hours  each,  if  6  men  mow  24  acres  in  4  days  of  9  hours  each  ? 

11.  If  4  lb.  6  oz.  of  tea  cost  $  2^-^,  what  will  3  lb.  11  oz.  cost  ? 

12.  If  sufiicient  flour  to  fill  8  bags  containing  98  lb.  each  can  be 
produced  from  16  bushels  of  wheat,  how  many  bushels  will  be  needed 
to  fill  14  barrels  of  196  lb.  each  ? 

13.  My  gas  bill  for  the  month  of  November  is  $  3.50  when  I  use 
6  burners  3^-  hours  each  evening.  How  much  ought  it  to  be  for  the 
month  of  December,  when  I  use  4  burners  for  5  hours  each  evening  ? 

14.  How  long  a  piece  of  cloth  .4  m.  wide  can  be  made  from  175  Kg. 
of  wool,  if  45  Kg.  make  a  piece  25  m.  long  and  .6  m.  wide  ? 

15.  How  many  hours  daily  ought  30  men  to  labor  to  perform  in  10 
days  a  piece  of  work  which  is  -|  as  great  as  a  similar  job  which  25 
men,  working  12  hours  per  day,  accomplished  in  12  days  ? 

16.  If  $  475  yield  %  171  interest  in  6  years,  how  long  will  it  take 
$  960  to  double  itself  at  the  same  rate  ? 

17.  A  bin  8  ft.  long,  6  ft.  wide,  and  41  ft.  deep  will  contain  270 
bushels  of  wheat.  How  deep  must  another  bin  be  built  that  is 
12  ft.  long  and  9  ft.  wide,  to  hold  405  bushels  ? 


APPENDIX 


405 


TOWNSHIP 


TOWNSHJP 


NORTH 


!S! 


NORTH 


BASE 


m — rn- 


LINE 


GOVERNMENT   LANDS 

The  government  lands  of  the  United  States  are  divided  by  par- 
allels and  meridians  into  townships,  6  miles  square.  Each  town- 
ship is  divided  into  36  square 
miles,  or  sections.  Each  section 
is  subdivided  into  half-sections 
and  quarter-sections. 

In  surveying  the  public  lands, 
lines  6  miles  apart  are  run  from 
east  to  west  and  from  north  to 
south,  dividing  the  territory  into 
square  townships.  An  east  and 
west  line  is  established  as  a  base 
line,  and  a  north  and  south  line 
as  a  principal  meridian. 

A  line  of  townships  running 
east  and  west  is  called  a  tier, 
and  a  line  of  townships  running  north  and  south  is  called  a  range. 

Any  township  is  designated  by  its  number  north  or  south  of  the 
base  line,  and  its  number  east  or  west  from  the  principal  meridian. 

Thus,  a  township  that  is  in  the  loth  tier  north  of  the  base  line, 
and  in  the  28th  range  east  of  the  4th  principal  meridian,  is  desig- 
nated :  T.  15  N.  E.  28  E.  4th  P.M. 

There  being  36  sections  in  a  township,  each  section  is  designated 
by  a  number.  The  numbering  begins  at  the  N.E.  corner,  increasing 
toward  the  west  and  east,  as  shown  in  the  accompanying  diagram. 


TOWNSHIP 


TOWNSHIP 


SOUTH 
SOUTH 


TOWNSHIP 

N 


W 


6 

5 

4 

3 

2 

1 

7 

8 

9 

lO 

11 

12 

18 

17 

16 

15 

14 

13 

19 

20 

21 

22 

23 

24 

30 

29 

28 

27 

26 

25 

31 

32 

33 

34 

35 

36 

SECTION 

N 
ONE  MILE 


w 


SIX  MILES 

8 


Ul 

320  A. 

ICO  A. 

111 

W.J^ 
'  of 
S.E.M 
80  A. 

of8.E.Ji 
40  A. 

40  A.    1 

ONE  MILE 


406  GRAMMAR  SCHOOL  ARITHMETIC  , 

GREATEST  COMMON  DIVISOR  BY  CONTINUED  DIVISION 
PRINCIPLES 

1.  A  divisor  of  a  number  will  divide  any  multiple  of  that  number. 

2.  A  common  divisor  of  two  numbers  will  divide  their  sum  and 
their  difference. 

1.  Find  the  G.  C.  D.  of  1395  and  1798. 


1395)1798  Any  common  divisor  of  1395  and  1798  will 

1395         3  divide  their  difference,  or  403.     Any  divisor  of 

-              ^  403  will  divide  3  times  403,  or  1209.    Any  cora- 

403)1395  mon  divisor  of  1395  and  1209  will  divide  their 

1209       2  difference,  or  31.     Therefore  the  G.  C.  D.  can- 

186H03  ^^^  ^®  greater  than  31.     By  a  similar  use  of  the 

^79       f\  principles  stated  above,  it  may  be  shovsrn  that  31 

£i^ r  will  divide  186,  372,  403,  1209,  1395,  and  1798. 

31)186  Hence  31  is  the  G.  C.  D.  of  1395  and  1798. 
186 

Summary 

To  find  the  greatest  common  divisor  of  two  numbers^  divide  the  greater  by  the 
less^  and  the  last  divisor  by  the  last  remainder,  continuing  the  process  until  there 
is  no  remainder.     The  divisor  last  used  is  the  greatest  common  divisor  required. 

When  more  than  two  7iumbers  are  given,  find  the  greatest  common  divisor  oj 
two  of  them;  then  of  that  greatest  common  divisor  and  one  of  the  remaining 
numbers,  and  so  on  till  all  of  the  numbers  have  been  used.  The  greatest  com- 
mon divisor  last  found  is  the  greatest  common  divisor  of  all  the  given  numbers. 

Find  the  G.  G.  D.  of 

2.  672  and  960  10.'  1650  and  1920 

3.  616  and  1012  11.  696, 1218,  and  1160 

4.  272  and  428  12.  450,  720,  and  810 

5.  1034  and  987  13.  465,  434,  and  341 

6.  1802  and  1431  14.  738,  553,  and  1271 

7.  2989  and  1830  15.  1316,  517,  and  1504 

8.  2263  and  3604  16.  1554,  2590,  and  703 

9.  5494  and  4355  17.  649,  2065,  and  2478 


APPENDIX 


407 


FARMERS'    ESTIMATES 
To  find  the  number  of  bushels  in  a  bin  or  granary, 
Divide  the  number  of  cubic  feet  in  the  bin  or  granary  by  1\, 

To  find  how  large  a  bin  will  contain  a  given  number  of  bushels, 

Multiply  the  number  of  bushels  by  1\. 

The  result  is  the  number  of  cubic  feet  in  the  required  bin. 

To  find  the  number  of  gallons  of  water  in  a  cistern  or  tank, 

Multiply  the  number  of  cubic  feet  of  water  by  7^. 

To  find  how  large  a  cistern  will  hold  a  given  number  of  gallons. 

Divide  the  number  of  gallons  by  7^. 

The  result  will  be  the  number  of  cubic  feet  in  the  required  cistern. 

To  find  how  many  bushels  of  shelled  corn  are  equal  to  a  given  number 
of  bushels  of  corn  in  the  ear, 

Divide  the  number  of  bushels  of  corn  in  the  ear  by  2. 

The  following  table  shows  the  number  of  pounds  in  a  legal  bushel, 
of  different  commodities,  in  various  states : 


Wheat 

Indian  Corn,  shelled  . 

Oats 

Barley 

Buckwheat  .... 

Rye 

Clover  Seed  .... 
Timothy  Seed  .  .  . 
Blue  Grass  Seed    .     . 


mm  (50  G4 

45 


48 
50 
5()5g!56 
'(>0 
45 


14 


Beans,  peas,  and  potatoes  usually  60  lb.  j  in  N.Y.,  beans  Qf2  lb. 


408  GRAMMAR  SCHOOL   ARITHMETIC 

Coal,  80  lb.,  except  Ind.,  70  or  80,  and  Ky.  76  lb. 

Salt :  111.,  50  lb.  common,  or  55  lb.  fine, 

N.J.,  56  lb.,  Ind.,  Ky.,  and  Iowa  50  lb., 

Pemi.,  80  lb.  coarse,  70  lb.  ground,  or  62  lb.  fine. 

KINDS  OF  PAPER  MONEY 

The  paper  money  of  this  country  is  of  four  kinds,  viz. : 

1.  United  States  Treasury  Notes. 

These  are  promises  of  the  United  States  to  pay  to  the  bearer,  on  demand,  the 
sum  named  in  the  note.  They  are  given  and  received  in  ordinary  business  trans- 
actions on  a  par  with  gold,  because  all  people  believe  that  the  United  States  is 
able  to  fulfill  its  promises  and  will  do  so. 

Treasury  notes  can  be  exchanged  for  gold  at  any  time,  but  people  prefer  the 
notes  for  most  purposes,  because  they  are  more  convenient  to  carry  and  less 
liable  to  be  lost.     Why  cannot  notes  of  individuals  be  used  for  money  ? 

2.  National  Bank  Notes. 

A  national  bank  note  is  a  promise  by  a  national  bank  to  pay  to  the  bearer,  on 
demand,  a  specified  sum  of  money.  Every  national  bank,  in  order  to  issue  this 
kind  of  money,  must  own  bonds  of  the  United  States  at  least  equal  in  amount 
to  the  notes  which  it  issues.  These  bonds,  although  owned  by  the  bank,  are 
held  by  the  Treasurer  of  the  United  States. 

If  any  national  bank  should  fail,  or  refuse  to  pay  its  notes,  the  United  States 
government  would  pay  them  and  take  its  payment  from  the  bonds  in  its  pos- 
session. So  that  the  credit  of  the  United  States  is  really  what  gives  value  to 
national  bank  notes. 

3.  Gold  Certificates. 

These  are  paper  bills  certifying  that  there  is  gold  on  deposit  in  the  United 
States  Treasury  of  a  value  corresponding  to  the  denomination  of  the  certificate, 
payable  to  the  bearer  of  the  certificate  on  demand. 

The  holder  of  the  certificate  may  exchange  it  for  gold  at  any  time.  The  value 
of  a  gold  certificate,  therefore,  depends  on  the  fact  that  there  is  an  amount  of 
gold  in  the  Treasury  designed  expressly  for  the  payment  of  the  certificate. 

4.  Silver  Certificates. 

These  are  similar  to  gold  certificates,  except  that  they  are  secured  by  silver 
instead  of  gold,  on  deposit  in  the  treasury. 

Ask  your  father  to  let  you  take  some  paper  money  to  examine.  See  if  you 
can  tell  to  which  class  of  paper  money  it  belongs,  and  upon  what  its  value 
depends. 


THE  MULTIPLICATION   TABLE 


409 


2x    1=   2 

3x    1=   3 

4x    1=   4 

5x    1=   5 

2x    2=    4 

3x2=6 

4x    2=    8 

5x    2  =  10 

2x    3=    6 

3x    3=    9 

4x    3  =  12 

5x    3  =  15 

2x    4=    8 

3x    4  =  12 

4  X    4  =  16 

5x    4  =  20 

2x    5  =  10 

3x    5  =  15 

4x    5  =  20 

5x    5  =  25 

2x    6  =  12 

3x    6  =  18 

4x    6  =  24 

5x    6  =  30 

2x    7  =  14 

3x    7  =  21 

4x    7  =  28 

5x    7  =  35 

2x    8  =  16 

3x    8  =  24 

4x    8  =  32 

5x    8  =  40 

2x    9  =  18 

3  X    9  =  27 

4x    9  =  36 

5x    9  =  45 

2  X  10  =  20 

3  X  10  =  30 

4  X  10  =  40 

5  X  10  =  50 

2  x  11  =  22 

3  X  11  =  33 

4  X  11  =  44 

5  X  11  =  55 

2  X  12  =  24 

3  X  12  =  36 

4  X  12  =  48 

5  X  12  =  60 

6X    1=    6 

7x    1=    7 

8x    1=   8 

9x1=     9 

6x    2  =  12 

7x    2  =  14 

8  X    2  =  16 

9  X    2  =    18 

6x    3  =  18 

7x    3  =  21 

8x    3  =  24 

9x    3=    27 

6x    4  =  24 

7x    4  =  28 

8x    4  =  32 

9  X    4  =   36 

6x    5  =  30 

7X    5  =  35 

8  X    5  =  40 

9x    5=   45 

6  X    6  =  36 

7x    6  =  42 

8x    6  =  48 

9x    6=    54 

6x    7  =  42 

7  X    7  =  49 

8x    7  =  56 

9x    7=    63 

6x    8  =  48 

7x    8  =  56 

8x    8  =  64 

9x    8=    72 

6  X    9  =  54 

7  X    9  =  63 

8x    9  =  72 

9  X    9  =    81 

6  X  10  =  60 

7  x  10  =  70 

8  X  10  =  80 

9x10=    90 

6  X  11  =  66 

7  X  11  =  77 

8  x  11  =  88 

9x11=    99 

6  X  12  =  72 

7  X  12  =  84 

8  X  12  =  96 

9x12  =  108 

10  X    1=    10 

11  X    1  =   11 

12  X    1=    12 

KOMAN 

10  X    2=    20 

11  X    2=    22 

12  X    2=    24 

Numerals 

10  X    3=    30 

11  X    3  =    33 

12  X    3  =   36 

I  =1 

10  X    4=    40 

11  X    4  =   44 

12  X    4=    48 

10  X    5=    50 

11  X    5=    55 

12  X    5=    60 

V=5 

10  X    6  =    60 

11  X    6  =    66 

12  X    6  =    72 

X=10 

10  X    7=    70 

11  X    7  =   77 

12  X    7  =    84 

L  =50 

10  X    8=    80 

11  X    8=    88 

12  X    8  =    96 

C  =100 

10  X    9=    90 

11  X    9  =    99 

12  X    9  =  108 

D=500 

10  X  10  =  100 

11  X  10  =  110 

12  X  10  =  120 

10  X  11  =  110 

11  X  11  =  121 

12  X  11  =  132 

M  =  1000 

10x12  =  120 

11  X  12  =  132 

12  X  12  =  144 

M  =  1,000,000 

410 


GRAMMAR  SCHOOL   ARITHMETIC 


Compound  Interest  Table 


Periods 

%  Per  Cent 

1  Per  Cent 

li/i  Per  Cent 

11/2  Per  Cent 

2  Per  Cent 

2V2  Per  Cent 

1 

1.007500 

1.010000 

1.012500 

1.015000 

1.020000 

1.025000 

2 

1.015056 

1.020100 

1.025156 

1.030225 

1.040400 

1.050625 

3 

1.022669 

1.030301 

1.037970 

1.045678 

1.061208 

1.076891 

4 

1.030339 

1.040604 

1.050945 

1.061364 

1.082432 

1.103813 

5 

1.038066 

1.051010 

1.064082 

1.077284 

1.104981 

1.131408 

6 

1.045852 

1.061520 

1.077383 

1.093443 

1.126162 

1.159693 

7 

1.053696 

1.072135 

1.090850 

1.109845 

1.148686 

1.188686 

8 

1.061598 

1.082856 

1.104486 

1.126493 

1.171660 

1.218403 

9 

1.069560 

1.093685 

1.118292 

1.143390 

1.195093 

1.248863 

10 

1.077582 

1.104622 

1.132270 

1.160541 

1.218994 

1.280085 

11 

1.085664 

1.115668 

1.146424 

1.177949 

1.243374 

1.312087 

12 

1.093806 

1.126825 

1.160754 

1.195618 

1.268242 

1.344889 

13 

1.103010 

1.138093 

1.175263 

1.213552 

1.293607 

1.378511 

14 

1.110275 

1.149474 

1.189954 

1.231756 

1.319479 

1.412774 

15 

1.118602 

1.160968 

1.204829 

1.250232 

1.345868 

1.448298 

16 

1.126992 

1.172578 

1.219889 

1.268985 

1.372786 

1.484506 

17 

1.135444 

1.184304 

1.235138 

1.288020 

1.400241 

1.521618 

18 

1.143960 

1.196147 

1.250477 

1.307341 

1.428246 

1.559659 

19 

1.152540 

1.208108 

1.266108 

1.326951 

1.456811 

1.598650 

20 

1.161184 

1.220190 

1.281934 

1.346855 

1.485947 

1.638616 

Periods 

3  Per  Cent 

314  Per  Cent 

4  Per  Cent 

5  Per  Cent 

6  Per  Cent 

7  Per  Cent 

1 

1.030000 

1.035000 

1.040000 

1.050000 

1.060000 

1.070000 

2 

1.060900 

1.071225 

1.081600 

1.102500 

1.123600 

1.144900 

3 

1.092727 

1.108718 

1.124864 

1.157625 

1.191016 

1.225043 

4 

1.125509 

1.147523 

1.169859 

1.215506 

1.262477 

1.310796 

5 

1.159274 

1.187686 

1.216653 

1.276282 

1.338226 

1.402552 

6 

1.194052 

1.229255 

1.265319 

1.340096 

1.418519 

1.500730 

7 

1.229874 

1.272279 

1.315932 

1.407100 

1.503630 

1.605781 

8 

1.266770 

1.316809 

1.368569 

1.477455 

1.593848 

1.718186 

9 

1.304773 

1.362897 

1.423312 

1.551328 

1.689479 

1.838459 

10 

1.343916 

1.410599 

1.480244 

1.628895 

1.790848 

1.967151 

11 

1.384234 

1.459970 

1.539454 

1.710339 

1.898299 

2.104852 

12 

1.425761 

1.511069 

1.601032 

1.795856 

2.012197 

2.252192 

13 

1.468534 

1.563956 

1.665074 

1.885649 

2.132928 

2.409845 

14 

1.512590 

1.618695 

1.731676 

1.979932 

2.260904 

2.578534 

15 

1.557967 

1.675349 

1.800944 

2.078928 

2.396558 

2.759031 

16 

1.604706 

1.733986 

1.872981 

2.182875 

2.540352 

2.952164 

17 

1.652848 

1.794676 

1.947901 

2.292018 

2.692773 

3.158815 

18 

1.702433 

1.857489 

2.025817 

2.406619 

2.854339 

3.379932 

19 

1.753506 

1.922501 

2.106849 

2.526950 

3.025600 

3.616527 

20 

1.806111 

1.989789 

2.191123 

2.653298 

3.207136 

3.869684 

INDEX 


Abstract  number,  3. 
Acceptance,  233. 
Accounts,  61. 
Acute  angle,  83. 
Addends,  10. 
Addition,  10. 

of  compound  numbers,  91. 

of  fractions  and  mixed  numbers,  42. 
Ad  valorem  dutj^,  262. 
Agent,  145. 
Aliquot  parts,  55. 
Altitude,  96. 

of  a  cone,  356. 

of  a  regular  pyramid,  358. 
Amount,  129, 165,  191. 
Angle,  81. 
Antecedent,  303. 

Applications  of  square  root,  340. 
Arabic  notation,  4. 
Arc,  81. 
Areas  of  parallelograms,  98. 

of  rectangles,  97. 

of  regular  polygons,  345. 

of  trapezoids,  346. 

of  triangles,  99. 
Articles  sold  by  the  100,  etc.,  75. 
Assessment,  288. 
Assessment  roll,  221. 
Assessors,  221. 
Axioms,  270. 

Balance,  61. 

Bank  discount,  211. 

Bank  draft,  226. 

Bank  note,  211. 

Banks  and  banking,  205. 

savings,  205. 

of  deposit,  205. 

national,  206. 

state,  206. 
Base,  96,  129,  351, 


Base  line,  405. 
Bill,  62. 

Bonds,  296,  299. 
Braces,  23. 
Brackets,  23. 
Broker,  289. 
Brokerage,  145,  289. 
Building  walls,  102. 

Cable  transfers,  242. 

Cancellation,  31. 

Capacity,  113. 

Capital  stock,  287. 

Carat,  84. 

Certificate  of  stock,  284. 

Check,  207. 

Circle,  81,  347. 

Circumference,  81,  360,  347. 

Clearing  house,  229. 

Commercial  discount,  151. 

Commercial  drafts,  231. 

Commission,  145. 

Common  denominator,  41. 

Common  divisor,  34. 

Common  fraction,  51. 

Common  fraction  at  the  end  of  a  decimal, 
54. 

Common  multiple,  32. 

Common  stock,  289. 

Comparative  study  of  decimals  and  com- 
mon fractions,  51. 

Complex  fraction,  49. 

Composite  number,  29. 

Compound  fraction,  4(>. 

Compound  interest,  180. 

Compound  interest  table,  410. 

Compound  number,  76. 

Computation  in  hundredths,  125. 

Concrete  number,  3. 

Cone,  355. 

Consequent,  303. 

411 


412 


INDEX 


Consignee,  146. 
Consignment,  145. 
Consignor,  146. 
Contents,  100. 
Contract,  157. 
Corporation,  296. 
Correspondent,  228. 
Couplet,  304. 
Coupon,  299. 
Coupon  bonds,  299. 
Creditor,  62. 
Cube,  100,  324. 
Cube  root,  327,  392. 
Cylinder,  352. 

Day  of  discount,  212. 
Debit,  61. 
Debtor,  62. 
Decimal  fraction,  4. 
Default  of  payment,  188. 
Denominate  number,  76. 
Denomination,  76. 
Denominator,  3,  37. 
Diameter,  347,  359. 
Difference,  12,  129. 
Digit,  25. 
Direct  ratio,  304. 
Discount,  211,288. 
Dividend,  17,  288. 
Division,  17. 

of  compound  numbers,  95. 

of  decimals,  19. 

of  fractions,  48. 
Divisor,  17. 

Domestic  exchange,  229. 
Draft,  226. 
Drawee  of  a  check,  208. 

of  a  draft,  226. 
Drawer  of  a  check,  208. 

of  a  draft,  226. 
Duties  or  customs,  261. 

Equation,  267. 
Even  number,  25. 
Evolution,  328. 

by  factoring,  339. 
Exact  differences  between  dates,  94. 
Exact  interest,  171. 
Exchange,  228,  229,  233. 


Exponent,  324. 

Express  money  order,  237. 

Extremes,  306. 

Face  of  a  bond,  299. 

of  a  check,  208. 

of  a  draft,  226. 

of  a  note,  185. 

of  an  insurance  policy,  158. 
Factors,  15,  29. 
Farmers'  estimates,  407. 
Fathom,  83. 
Floor  covering,  104. 
Fluid  ounce,  77. 
Foreign  exchange,  239. 
Fraction,  3,  36. 
Franc,  240. 
Furlong,  83. 

Government  lands,  405. 
Greatest  common  divisor,  34. 
by  continued  division,  406. 
Guide  figure  in  division,  18. 

Hand,  83. 
Heptagon,  345. 
Hexagon,  345. 
Holder  of  a  note,  185. 
Hypotenuse,  340. 

Ideas  of  proportion,  27. 
Improper  fraction,  39. 
Indorsee,  187. 
Indorser,  187. 
Indorsement,  186. 

in  blank,  186. 

in  full,  187. 

of  partial  payments,  193. 

restrictive,  187. 
Insurance,  157. 
Integer,  3. 
Integral  factor,  29. 
Interest,  165. 

compound,  180. 

exact,  171. 

for  short  periods,  170. 

problems  in,  172. 

simple,  180. 
International  date  line,  367. 


INDEX 


413 


Intrinsic  par  of  exchange,  241. 
Inverse  ratio,  304. 
Invoice,  62. 
Involution,  325. 

Karat,  84. 

Kinds  of  paper  money,  408. 

Knot,  83. 

Lateral  surface  of  a  pyramid,  358. 
Least  common  denominator,  41. 
Least  common  multiple,  33. 
Legal  rate,  165. 
Legs  of  a  right  triangle,  340. 
Lira,  240. 
List  price,  151. 
Longitude  and  time,  365. 

Maker  of  a  note,  185. 

Mark,  240. 

Market  value,  288. 

Maturity,  187. 

Means,  306. 

Mensuration,  344. 

Meridian,  365. 

Methods  of  computing  interest,  397. 

bankers'  method,  398. 

by  aliquot  parts,  397. 

ordinary  six  per  cent  method,  398. 
Metric  system,  247. 
Minuend,  12. 
Mixed  decimal,  5. 
Mixed  number,  39. 
Multiple,  29. 
Multiplicand,  15. 
Multiplier,  15. 
Multiplication,  15. 

of  compound  numbers,  94. 

of  decimals,  19. 

of  fractions,  45. 

table,  409. 

Nautical  mile,  83. 
Net  price,  151. 
Net  proceeds,  146,  234. 
Notation,  4. 
Notes,  182. 

kinds  of,  185. 
Number,  3. 


Numbers  prime  to  each  other,  34. 
Numeration,  6. 
Numerator,  3,  37. 

Obtuse  angle,  83. 

Octagon,  345. 

Odd  number,  26. 

Of  between  fractions,  46. 

Orders  of  units,  4. 

Parallel  lines,  96. 
Parallelogram,  96. 
Partial  payments,  193. 
Parties,  61. 

Partitive  proportion,  312. 
Partnership,  314. 
Par  value,  288. 
Payee  of  a  check,  208. 

of  a  draft,  226. 

of  a  note,  185. 
Pentagon,  345. 
Percentage,  128,  129. 
Per  cents  equivalent  to  common  fractions, 

133. 
Perch,  84. 
Perfect  cube,  328. 
Perfect  power,  328. 
Period,  4,  5. 
Perpendicular,  340. 
Personal  property,  220. 
Plane  figure,  345. 
Plane  surface,  345. 
Policy,  158. 
Poll  tax,  220. 
Polygon,  345. 
Postal  money  order,  235. 
Power,  3,  324. 
Preferred  stock,  289. 
Premium,  158,  288. 
Present  worth,  398. 
Prime  factor,  29. 
Prime  meridian,  365. 
Prime  number,  29. 
Principal,  146, 165,  191. 
Prism,  351. 

Proceeds  of  a  note,  211. 
Product,  15. 
Profit  and  loss,  139. 
Proper  fraction,  39. 


414 


INDEX 


Property  tax,  220. 
Proportion,  306. 
Protest,  218. 

Quadrilateral,  96. 
Quotient,  17. 

Radical  index,  327. 
Radical  sign,  327. 
Radius,  347,  359. 
Rate  of  interest,  165. 
Rate  per  cent,  129. 
Ratio,  303. 
Real  property,  220. 
Rectangle,  96. 
Rectangular  prism,  351. 
Reduction,  37. 
ascending,  85. 
descending,  85. 

of  a  fraction  to  lowest  terms,  37. 
of  complex   fractions  to  simple  frac- 
tions, 49. 
of  fractions  to  least  common  denomi- 
nator, 41. 
of  improper  fractions  to  integers  or 

mixed  numbers,  39. 
of  integers  and  mixed  numbers  to  im- 
proper fractions,  40. 
Registered  bonds,  299. 
Regular  polygon,  345. 
Regular  pyramid,  358. 
Remainder,  12,  17. 
Review  and  practice,  68-74,  117-125,  198- 

205,  317-323,  373-391. 
Right  triangle,  340. 
Roman  notation,  8. 
Root,  327. 
Rules 

for  finding  whether  a  number  is  prime 

or  composite,  29. 
for  finding  the  number  of  board  feet, 

109. 
for  finding  bank   discount  and  pro- 
ceeds, 212. 
for  partial  payments,  193. 
Merchants'  rule,  197. 

Scale  of  Arabic  notation,  4. 
Section  of  land,  84,  405. 


Share,  287. 

Short  division,  18. 

Sight  draft,  233. 

Sign  of  equality,  10. 

Significant  figures,  4. 

Signs  of  aggregation,  23. 

Similar  solids,  397. 

Similar  surfaces,  361. 

Simple  fraction,  49. 

Simple  interest,  180. 

Simple  number,  76. 

Simplest  form  of  a  number,  42. 

Slant  height  of  a  cone,  356. 

of  a  pyramid,  358. 
Solid,  351. 
Sovereign,  239. 
Special  cases  in  division,  58. 
Special  cases  in  multiplication,  57. 
Specific  duty,  262. 
Sphere,  359. 
Square,  324. 

Square  of  roofing,  etc.,  84. 
Square  prism,  351. 
Square  root,  327,  329. 

of  a  common  fraction,  337. 

of  a  decimal,  336. 
Standard  time,  371. 
Statement,  62. 
Statute  mile,  83. 
Stock  company,  287. 
Stockholder,  287. 
Subtraction,  12. 

of  compound  numbers,  91. 

of  fractions  and  mixed  numbers,  43. 
Subtrahend,  12. 
Successive  discounts,  151. 
Sum,  10. 
Suretyship,  400. 

Table  of 

apothecaries'  weight,  77. 
Arabic  notation,  5. 
arc  and  angle  measure,  81. 
avoirdupois  weight,  76. 
compound  interest,  210. 
counting,  78. 
dry  measure,  76. 
English  money,  80. 
French  money,  80. 


INDEX 


415 


Table  of 

German  money,  81. 

linear  measure,  77. 

liquid  measure,  76. 

paper  measure,  79. 

surface  measure,  78. 

surveyors'  long  measure,  78, 

surveyors'  square  measure,  78. 

time,  79. 

Troy  weight,  77. 

United  States  money,  80. 

volume  measure,  78. 
Tare,  263. 
Tariff,  262. 
Tax,  220. 
Tax  budget,  221. 
Tax  rate,  221. 
Term  of  discount,  212. 
Telegraph  money  order,  238. 


Terms  of  a  fraction,  37. 

of  a  ratio,  303. 

of  a  proportion,  306. 
Tests  of  divisibility,  25. 
Time  draft,  233. 
Trade  discount,  151. 
True  discount,  398. 
Trust  companies,  206. 

Usury,  165. 

Value  of  a  fraction,  37. 
Vertex  of  a  cone,  355. 
Vinculum,  23. 
Volume,  100,  113. 

of  a  cone,  357. 

of  a  cylinder,  354. 

of  a  prism,  351, 

of  a  rectangular  prism,  351. 

of  a  pyramid,  358. 


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